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Properties

Label 2-294-7.4-c5-0-9
Degree 22
Conductor 294294
Sign −0.947+0.318i-0.947 + 0.318i
Analytic cond. 47.152847.1528
Root an. cond. 6.866796.86679
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  + (2 + 3.46i)2-s + (4.5 − 7.79i)3-s + (−7.99 + 13.8i)4-s + (51.7 + 89.6i)5-s + 36·6-s − 63.9·8-s + (−40.5 − 70.1i)9-s + (−206. + 358. i)10-s + (−120. + 208. i)11-s + (72 + 124. i)12-s − 805.·13-s + 931.·15-s + (−128 − 221. i)16-s + (−646. + 1.12e3i)17-s + (162 − 280. i)18-s + (−137. − 238. i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.925 + 1.60i)5-s + 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.654 + 1.13i)10-s + (−0.299 + 0.518i)11-s + (0.144 + 0.249i)12-s − 1.32·13-s + 1.06·15-s + (−0.125 − 0.216i)16-s + (−0.542 + 0.939i)17-s + (0.117 − 0.204i)18-s + (−0.0875 − 0.151i)19-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=((−0.947+0.318i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+5/2) L(s)=((−0.947+0.318i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: −0.947+0.318i-0.947 + 0.318i
Analytic conductor: 47.152847.1528
Root analytic conductor: 6.866796.86679
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ294(67,⋅)\chi_{294} (67, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 294, ( :5/2), −0.947+0.318i)(2,\ 294,\ (\ :5/2),\ -0.947 + 0.318i)

Particular Values

L(3)L(3) ≈\approx 1.4445893781.444589378
L(12)L(\frac12) ≈\approx 1.4445893781.444589378
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)
bad2 1+(−2−3.46i)T 1 + (-2 - 3.46i)T
3 1+(−4.5+7.79i)T 1 + (-4.5 + 7.79i)T
7 1 1
good5 1+(−51.7−89.6i)T+(−1.56e3+2.70e3i)T2 1 + (-51.7 - 89.6i)T + (-1.56e3 + 2.70e3i)T^{2}
11 1+(120.−208.i)T+(−8.05e4−1.39e5i)T2 1 + (120. - 208. i)T + (-8.05e4 - 1.39e5i)T^{2}
13 1+805.T+3.71e5T2 1 + 805.T + 3.71e5T^{2}
17 1+(646.−1.12e3i)T+(−7.09e5−1.22e6i)T2 1 + (646. - 1.12e3i)T + (-7.09e5 - 1.22e6i)T^{2}
19 1+(137.+238.i)T+(−1.23e6+2.14e6i)T2 1 + (137. + 238. i)T + (-1.23e6 + 2.14e6i)T^{2}
23 1+(1.89e3+3.28e3i)T+(−3.21e6+5.57e6i)T2 1 + (1.89e3 + 3.28e3i)T + (-3.21e6 + 5.57e6i)T^{2}
29 1−1.22e3T+2.05e7T2 1 - 1.22e3T + 2.05e7T^{2}
31 1+(−2.81e3+4.87e3i)T+(−1.43e7−2.47e7i)T2 1 + (-2.81e3 + 4.87e3i)T + (-1.43e7 - 2.47e7i)T^{2}
37 1+(−4.53e3−7.86e3i)T+(−3.46e7+6.00e7i)T2 1 + (-4.53e3 - 7.86e3i)T + (-3.46e7 + 6.00e7i)T^{2}
41 1−1.82e4T+1.15e8T2 1 - 1.82e4T + 1.15e8T^{2}
43 1+1.17e4T+1.47e8T2 1 + 1.17e4T + 1.47e8T^{2}
47 1+(1.15e4+1.99e4i)T+(−1.14e8+1.98e8i)T2 1 + (1.15e4 + 1.99e4i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+(8.83e3−1.52e4i)T+(−2.09e8−3.62e8i)T2 1 + (8.83e3 - 1.52e4i)T + (-2.09e8 - 3.62e8i)T^{2}
59 1+(−9.18e3+1.59e4i)T+(−3.57e8−6.19e8i)T2 1 + (-9.18e3 + 1.59e4i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(−5.66e3−9.80e3i)T+(−4.22e8+7.31e8i)T2 1 + (-5.66e3 - 9.80e3i)T + (-4.22e8 + 7.31e8i)T^{2}
67 1+(1.80e4−3.12e4i)T+(−6.75e8−1.16e9i)T2 1 + (1.80e4 - 3.12e4i)T + (-6.75e8 - 1.16e9i)T^{2}
71 1+6.34e4T+1.80e9T2 1 + 6.34e4T + 1.80e9T^{2}
73 1+(2.64e4−4.58e4i)T+(−1.03e9−1.79e9i)T2 1 + (2.64e4 - 4.58e4i)T + (-1.03e9 - 1.79e9i)T^{2}
79 1+(−2.42e4−4.20e4i)T+(−1.53e9+2.66e9i)T2 1 + (-2.42e4 - 4.20e4i)T + (-1.53e9 + 2.66e9i)T^{2}
83 1+1.13e5T+3.93e9T2 1 + 1.13e5T + 3.93e9T^{2}
89 1+(−5.41e4−9.38e4i)T+(−2.79e9+4.83e9i)T2 1 + (-5.41e4 - 9.38e4i)T + (-2.79e9 + 4.83e9i)T^{2}
97 1+9.96e4T+8.58e9T2 1 + 9.96e4T + 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

−11.50507166395273522856383458769, −10.29671552771805619636232459263, −9.744912345085819228565626336094, −8.305289134725145891190615313530, −7.29433711899035757315862484277, −6.61545259779112006506340454457, −5.86227944470570397242242344409, −4.38179243061646611440402596918, −2.79476834306928668934256448948, −2.15926633740515517148271558999, 0.29408986605787766245766384009, 1.67471991784309015324619986436, 2.81591379628559282954867351180, 4.42972253595802301193796804279, 5.06355014619340073476629186913, 5.94286353236680425702889475061, 7.76098142906066108877135266947, 8.924479093612067559738689628410, 9.512456908400316509229969372022, 10.17936909605998170693083558377

Graph of the ZZ-function along the critical line