↑

Properties

Label 2-74-37.7-c7-0-20
Degree 22
Conductor 7474
Sign −0.819+0.573i-0.819 + 0.573i
Analytic cond. 23.116423.1164
Root an. cond. 4.807964.80796
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (7.51 − 2.73i)2-s + (28.9 + 10.5i)3-s + (49.0 − 41.1i)4-s + (−48.7 − 276. i)5-s + 246.·6-s + (−219. − 1.24e3i)7-s + (256. − 443. i)8-s + (−947. − 795. i)9-s + (−1.12e3 − 1.94e3i)10-s + (−3.01e3 + 5.22e3i)11-s + (1.85e3 − 674. i)12-s + (−7.24e3 + 6.07e3i)13-s + (−5.05e3 − 8.75e3i)14-s + (1.50e3 − 8.52e3i)15-s + (711. − 4.03e3i)16-s + (−210. − 176. i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (0.619 + 0.225i)3-s + (0.383 − 0.321i)4-s + (−0.174 − 0.989i)5-s + 0.465·6-s + (−0.241 − 1.37i)7-s + (0.176 − 0.306i)8-s + (−0.433 − 0.363i)9-s + (−0.355 − 0.615i)10-s + (−0.683 + 1.18i)11-s + (0.309 − 0.112i)12-s + (−0.914 + 0.766i)13-s + (−0.492 − 0.852i)14-s + (0.114 − 0.651i)15-s + (0.0434 − 0.246i)16-s + (−0.0103 − 0.00870i)17-s + ⋯

Functional equation

Λ(s)=(74s/2 ΓC(s) L(s)=((−0.819+0.573i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(74s/2 ΓC(s+7/2) L(s)=((−0.819+0.573i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7474    =    2⋅372 \cdot 37
Sign: −0.819+0.573i-0.819 + 0.573i
Analytic conductor: 23.116423.1164
Root analytic conductor: 4.807964.80796
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ74(7,⋅)\chi_{74} (7, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 74, ( :7/2), −0.819+0.573i)(2,\ 74,\ (\ :7/2),\ -0.819 + 0.573i)

Particular Values

L(4)L(4) ≈\approx 0.643199−2.04169i0.643199 - 2.04169i
L(12)L(\frac12) ≈\approx 0.643199−2.04169i0.643199 - 2.04169i
L(92)L(\frac{9}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−7.51+2.73i)T 1 + (-7.51 + 2.73i)T
37 1+(−6.25e4+3.01e5i)T 1 + (-6.25e4 + 3.01e5i)T
good3 1+(−28.9−10.5i)T+(1.67e3+1.40e3i)T2 1 + (-28.9 - 10.5i)T + (1.67e3 + 1.40e3i)T^{2}
5 1+(48.7+276.i)T+(−7.34e4+2.67e4i)T2 1 + (48.7 + 276. i)T + (-7.34e4 + 2.67e4i)T^{2}
7 1+(219.+1.24e3i)T+(−7.73e5+2.81e5i)T2 1 + (219. + 1.24e3i)T + (-7.73e5 + 2.81e5i)T^{2}
11 1+(3.01e3−5.22e3i)T+(−9.74e6−1.68e7i)T2 1 + (3.01e3 - 5.22e3i)T + (-9.74e6 - 1.68e7i)T^{2}
13 1+(7.24e3−6.07e3i)T+(1.08e7−6.17e7i)T2 1 + (7.24e3 - 6.07e3i)T + (1.08e7 - 6.17e7i)T^{2}
17 1+(210.+176.i)T+(7.12e7+4.04e8i)T2 1 + (210. + 176. i)T + (7.12e7 + 4.04e8i)T^{2}
19 1+(1.87e4+6.82e3i)T+(6.84e8+5.74e8i)T2 1 + (1.87e4 + 6.82e3i)T + (6.84e8 + 5.74e8i)T^{2}
23 1+(5.09e3+8.82e3i)T+(−1.70e9+2.94e9i)T2 1 + (5.09e3 + 8.82e3i)T + (-1.70e9 + 2.94e9i)T^{2}
29 1+(−6.21e4+1.07e5i)T+(−8.62e9−1.49e10i)T2 1 + (-6.21e4 + 1.07e5i)T + (-8.62e9 - 1.49e10i)T^{2}
31 1−2.52e5T+2.75e10T2 1 - 2.52e5T + 2.75e10T^{2}
41 1+(−2.71e4+2.28e4i)T+(3.38e10−1.91e11i)T2 1 + (-2.71e4 + 2.28e4i)T + (3.38e10 - 1.91e11i)T^{2}
43 1−1.91e5T+2.71e11T2 1 - 1.91e5T + 2.71e11T^{2}
47 1+(3.11e5+5.40e5i)T+(−2.53e11+4.38e11i)T2 1 + (3.11e5 + 5.40e5i)T + (-2.53e11 + 4.38e11i)T^{2}
53 1+(−6.97e4+3.95e5i)T+(−1.10e12−4.01e11i)T2 1 + (-6.97e4 + 3.95e5i)T + (-1.10e12 - 4.01e11i)T^{2}
59 1+(5.55e4−3.15e5i)T+(−2.33e12−8.51e11i)T2 1 + (5.55e4 - 3.15e5i)T + (-2.33e12 - 8.51e11i)T^{2}
61 1+(−1.57e6+1.31e6i)T+(5.45e11−3.09e12i)T2 1 + (-1.57e6 + 1.31e6i)T + (5.45e11 - 3.09e12i)T^{2}
67 1+(3.54e5+2.00e6i)T+(−5.69e12+2.07e12i)T2 1 + (3.54e5 + 2.00e6i)T + (-5.69e12 + 2.07e12i)T^{2}
71 1+(1.13e6+4.13e5i)T+(6.96e12+5.84e12i)T2 1 + (1.13e6 + 4.13e5i)T + (6.96e12 + 5.84e12i)T^{2}
73 1−7.81e4T+1.10e13T2 1 - 7.81e4T + 1.10e13T^{2}
79 1+(−5.40e5−3.06e6i)T+(−1.80e13+6.56e12i)T2 1 + (-5.40e5 - 3.06e6i)T + (-1.80e13 + 6.56e12i)T^{2}
83 1+(−6.99e6−5.86e6i)T+(4.71e12+2.67e13i)T2 1 + (-6.99e6 - 5.86e6i)T + (4.71e12 + 2.67e13i)T^{2}
89 1+(−1.44e6+8.19e6i)T+(−4.15e13−1.51e13i)T2 1 + (-1.44e6 + 8.19e6i)T + (-4.15e13 - 1.51e13i)T^{2}
97 1+(5.47e6+9.47e6i)T+(−4.03e13+6.99e13i)T2 1 + (5.47e6 + 9.47e6i)T + (-4.03e13 + 6.99e13i)T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−12.78824459021682165343058731033, −11.88222647393360601950381800350, −10.33632846482851448589544533417, −9.444498483197817081348599131030, −7.994696613079853464152768149864, −6.74648601325724013530154458415, −4.78748644158938145720774538116, −4.06824325209370860356476096366, −2.36005804166794176646753151997, −0.50260738162946491454302416117, 2.66556439689707884125834754664, 2.96993060421344279609906898525, 5.26142299351763540120532101247, 6.31469848878554150475007376231, 7.81077694512016852692988918730, 8.655587623139364286921253295235, 10.40766918204209383892463113999, 11.52450671829523774937836840176, 12.65092677452223669752522962480, 13.70440765399936530678362079713

Graph of the ZZ-function along the critical line