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Properties

Label 2-63-7.2-c5-0-14
Degree 22
Conductor 6363
Sign −0.984−0.174i-0.984 - 0.174i
Analytic cond. 10.104110.1041
Root an. cond. 3.178703.17870
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.11 − 8.85i)2-s + (−36.2 − 62.8i)4-s + (11.8 − 20.5i)5-s + (30.6 − 125. i)7-s − 414.·8-s + (−121. − 210. i)10-s + (232. + 403. i)11-s − 1.01e3·13-s + (−958. − 915. i)14-s + (−957. + 1.65e3i)16-s + (280. + 486. i)17-s + (693. − 1.20e3i)19-s − 1.72e3·20-s + 4.75e3·22-s + (2.05e3 − 3.56e3i)23-s + ⋯
L(s)  = 1  + (0.903 − 1.56i)2-s + (−1.13 − 1.96i)4-s + (0.212 − 0.367i)5-s + (0.236 − 0.971i)7-s − 2.28·8-s + (−0.383 − 0.665i)10-s + (0.580 + 1.00i)11-s − 1.67·13-s + (−1.30 − 1.24i)14-s + (−0.935 + 1.61i)16-s + (0.235 + 0.408i)17-s + (0.440 − 0.763i)19-s − 0.963·20-s + 2.09·22-s + (0.810 − 1.40i)23-s + ⋯

Functional equation

Λ(s)=(63s/2 ΓC(s) L(s)=((−0.984−0.174i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.174i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(63s/2 ΓC(s+5/2) L(s)=((−0.984−0.174i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6363    =    32⋅73^{2} \cdot 7
Sign: −0.984−0.174i-0.984 - 0.174i
Analytic conductor: 10.104110.1041
Root analytic conductor: 3.178703.17870
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ63(37,⋅)\chi_{63} (37, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 63, ( :5/2), −0.984−0.174i)(2,\ 63,\ (\ :5/2),\ -0.984 - 0.174i)

Particular Values

L(3)L(3) ≈\approx 0.212439+2.41369i0.212439 + 2.41369i
L(12)L(\frac12) ≈\approx 0.212439+2.41369i0.212439 + 2.41369i
L(72)L(\frac{7}{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)
bad3 1 1
7 1+(−30.6+125.i)T 1 + (-30.6 + 125. i)T
good2 1+(−5.11+8.85i)T+(−16−27.7i)T2 1 + (-5.11 + 8.85i)T + (-16 - 27.7i)T^{2}
5 1+(−11.8+20.5i)T+(−1.56e3−2.70e3i)T2 1 + (-11.8 + 20.5i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(−232.−403.i)T+(−8.05e4+1.39e5i)T2 1 + (-232. - 403. i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1+1.01e3T+3.71e5T2 1 + 1.01e3T + 3.71e5T^{2}
17 1+(−280.−486.i)T+(−7.09e5+1.22e6i)T2 1 + (-280. - 486. i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(−693.+1.20e3i)T+(−1.23e6−2.14e6i)T2 1 + (-693. + 1.20e3i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(−2.05e3+3.56e3i)T+(−3.21e6−5.57e6i)T2 1 + (-2.05e3 + 3.56e3i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1−2.38e3T+2.05e7T2 1 - 2.38e3T + 2.05e7T^{2}
31 1+(1.47e3+2.55e3i)T+(−1.43e7+2.47e7i)T2 1 + (1.47e3 + 2.55e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(−4.95e3+8.58e3i)T+(−3.46e7−6.00e7i)T2 1 + (-4.95e3 + 8.58e3i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1−4.47e3T+1.15e8T2 1 - 4.47e3T + 1.15e8T^{2}
43 1−5.18e3T+1.47e8T2 1 - 5.18e3T + 1.47e8T^{2}
47 1+(1.56e3−2.70e3i)T+(−1.14e8−1.98e8i)T2 1 + (1.56e3 - 2.70e3i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(−570.−988.i)T+(−2.09e8+3.62e8i)T2 1 + (-570. - 988. i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(−1.37e4−2.38e4i)T+(−3.57e8+6.19e8i)T2 1 + (-1.37e4 - 2.38e4i)T + (-3.57e8 + 6.19e8i)T^{2}
61 1+(−1.05e4+1.82e4i)T+(−4.22e8−7.31e8i)T2 1 + (-1.05e4 + 1.82e4i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(−2.77e4−4.81e4i)T+(−6.75e8+1.16e9i)T2 1 + (-2.77e4 - 4.81e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1−6.07e3T+1.80e9T2 1 - 6.07e3T + 1.80e9T^{2}
73 1+(−8.38e3−1.45e4i)T+(−1.03e9+1.79e9i)T2 1 + (-8.38e3 - 1.45e4i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(−2.42e3+4.19e3i)T+(−1.53e9−2.66e9i)T2 1 + (-2.42e3 + 4.19e3i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1+6.01e4T+3.93e9T2 1 + 6.01e4T + 3.93e9T^{2}
89 1+(3.12e4−5.41e4i)T+(−2.79e9−4.83e9i)T2 1 + (3.12e4 - 5.41e4i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1+6.36e4T+8.58e9T2 1 + 6.36e4T + 8.58e9T^{2}
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   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

−12.99533326876130253629722119211, −12.42402010727372423110855778097, −11.24976790682755540012549660378, −10.18020752216039981552873494098, −9.338492493681443476343619835912, −7.14391194419255722606568844502, −5.04437905922265098265754137820, −4.22625444851309139243255449987, −2.44655853350267015015861280444, −0.900750391196867634479850628379, 3.09802947825728272826189930714, 4.94743286478410068285764423294, 5.89551399692766232195554371632, 7.10360531170098358735588529716, 8.273118453698472296274698501049, 9.529528061078928753942223640790, 11.66297364727200424223477070071, 12.60242287644634859809301097199, 13.98362986695719015257406247753, 14.52998048216707017132261711824

Graph of the ZZ-function along the critical line