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Properties

Label 2-294-7.4-c5-0-31
Degree 22
Conductor 294294
Sign −0.701+0.712i-0.701 + 0.712i
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 11

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 + (13 + 22.5i)5-s + 36·6-s − 63.9·8-s + (−40.5 − 70.1i)9-s + (−51.9 + 90.0i)10-s + (−332 + 575. i)11-s + (72 + 124. i)12-s − 318·13-s + 234·15-s + (−128 − 221. i)16-s + (791 − 1.37e3i)17-s + (162 − 280. i)18-s + (118 + 204. i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.232 + 0.402i)5-s + 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.164 + 0.284i)10-s + (−0.827 + 1.43i)11-s + (0.144 + 0.249i)12-s − 0.521·13-s + 0.268·15-s + (−0.125 − 0.216i)16-s + (0.663 − 1.14i)17-s + (0.117 − 0.204i)18-s + (0.0749 + 0.129i)19-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=((−0.701+0.712i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+5/2) L(s)=((−0.701+0.712i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: −0.701+0.712i-0.701 + 0.712i
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: 11
Selberg data: (2, 294, ( :5/2), −0.701+0.712i)(2,\ 294,\ (\ :5/2),\ -0.701 + 0.712i)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
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+(−13−22.5i)T+(−1.56e3+2.70e3i)T2 1 + (-13 - 22.5i)T + (-1.56e3 + 2.70e3i)T^{2}
11 1+(332−575.i)T+(−8.05e4−1.39e5i)T2 1 + (332 - 575. i)T + (-8.05e4 - 1.39e5i)T^{2}
13 1+318T+3.71e5T2 1 + 318T + 3.71e5T^{2}
17 1+(−791+1.37e3i)T+(−7.09e5−1.22e6i)T2 1 + (-791 + 1.37e3i)T + (-7.09e5 - 1.22e6i)T^{2}
19 1+(−118−204.i)T+(−1.23e6+2.14e6i)T2 1 + (-118 - 204. i)T + (-1.23e6 + 2.14e6i)T^{2}
23 1+(1.10e3+1.91e3i)T+(−3.21e6+5.57e6i)T2 1 + (1.10e3 + 1.91e3i)T + (-3.21e6 + 5.57e6i)T^{2}
29 1+4.95e3T+2.05e7T2 1 + 4.95e3T + 2.05e7T^{2}
31 1+(3.56e3−6.17e3i)T+(−1.43e7−2.47e7i)T2 1 + (3.56e3 - 6.17e3i)T + (-1.43e7 - 2.47e7i)T^{2}
37 1+(2.17e3+3.77e3i)T+(−3.46e7+6.00e7i)T2 1 + (2.17e3 + 3.77e3i)T + (-3.46e7 + 6.00e7i)T^{2}
41 1+1.05e4T+1.15e8T2 1 + 1.05e4T + 1.15e8T^{2}
43 1+8.45e3T+1.47e8T2 1 + 8.45e3T + 1.47e8T^{2}
47 1+(−2.67e3−4.63e3i)T+(−1.14e8+1.98e8i)T2 1 + (-2.67e3 - 4.63e3i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+(−1.66e4+2.88e4i)T+(−2.09e8−3.62e8i)T2 1 + (-1.66e4 + 2.88e4i)T + (-2.09e8 - 3.62e8i)T^{2}
59 1+(7.71e3−1.33e4i)T+(−3.57e8−6.19e8i)T2 1 + (7.71e3 - 1.33e4i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(1.83e4+3.18e4i)T+(−4.22e8+7.31e8i)T2 1 + (1.83e4 + 3.18e4i)T + (-4.22e8 + 7.31e8i)T^{2}
67 1+(2.04e4−3.54e4i)T+(−6.75e8−1.16e9i)T2 1 + (2.04e4 - 3.54e4i)T + (-6.75e8 - 1.16e9i)T^{2}
71 1+9.09e3T+1.80e9T2 1 + 9.09e3T + 1.80e9T^{2}
73 1+(3.67e4−6.36e4i)T+(−1.03e9−1.79e9i)T2 1 + (3.67e4 - 6.36e4i)T + (-1.03e9 - 1.79e9i)T^{2}
79 1+(4.47e4+7.74e4i)T+(−1.53e9+2.66e9i)T2 1 + (4.47e4 + 7.74e4i)T + (-1.53e9 + 2.66e9i)T^{2}
83 1−6.42e3T+3.93e9T2 1 - 6.42e3T + 3.93e9T^{2}
89 1+(6.13e4+1.06e5i)T+(−2.79e9+4.83e9i)T2 1 + (6.13e4 + 1.06e5i)T + (-2.79e9 + 4.83e9i)T^{2}
97 1+2.13e4T+8.58e9T2 1 + 2.13e4T + 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

−10.39141152760035293305931558576, −9.633333957790276435313780215183, −8.424102401585275350381456304763, −7.31588493554219713927533012591, −6.97855858514161837554408929979, −5.56266500442447780979036743548, −4.63493313928673683933317343002, −3.07888643606651900056499376836, −2.01251152983812618917090761397, 0, 1.58307967033365036755482440288, 2.99685912860729848402954950362, 3.89282050406028101637071705149, 5.27843921549101402256145001295, 5.85050155749428893350056930861, 7.66324191946719382054960701838, 8.623177262176663772063503631736, 9.525697903333520092780379414056, 10.44304000470597809147317703080

Graph of the ZZ-function along the critical line