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Properties

Label 2-294-7.2-c5-0-11
Degree 22
Conductor 294294
Sign 0.900−0.435i0.900 - 0.435i
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 + (2.25 − 3.89i)5-s + 36·6-s − 63.9·8-s + (−40.5 + 70.1i)9-s + (−9.00 − 15.5i)10-s + (58.0 + 100. i)11-s + (72 − 124. i)12-s + 85.4·13-s + 40.5·15-s + (−128 + 221. i)16-s + (16.6 + 28.8i)17-s + (162 + 280. i)18-s + (317. − 550. i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.0402 − 0.0697i)5-s + 0.408·6-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.0284 − 0.0493i)10-s + (0.144 + 0.250i)11-s + (0.144 − 0.249i)12-s + 0.140·13-s + 0.0465·15-s + (−0.125 + 0.216i)16-s + (0.0139 + 0.0241i)17-s + (0.117 + 0.204i)18-s + (0.201 − 0.349i)19-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=((0.900−0.435i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+5/2) L(s)=((0.900−0.435i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: 0.900−0.435i0.900 - 0.435i
Analytic conductor: 47.152847.1528
Root analytic conductor: 6.866796.86679
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ294(79,⋅)\chi_{294} (79, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 294, ( :5/2), 0.900−0.435i)(2,\ 294,\ (\ :5/2),\ 0.900 - 0.435i)

Particular Values

L(3)L(3) ≈\approx 2.4841656262.484165626
L(12)L(\frac12) ≈\approx 2.4841656262.484165626
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+(−2.25+3.89i)T+(−1.56e3−2.70e3i)T2 1 + (-2.25 + 3.89i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(−58.0−100.i)T+(−8.05e4+1.39e5i)T2 1 + (-58.0 - 100. i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1−85.4T+3.71e5T2 1 - 85.4T + 3.71e5T^{2}
17 1+(−16.6−28.8i)T+(−7.09e5+1.22e6i)T2 1 + (-16.6 - 28.8i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(−317.+550.i)T+(−1.23e6−2.14e6i)T2 1 + (-317. + 550. i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(1.36e3−2.36e3i)T+(−3.21e6−5.57e6i)T2 1 + (1.36e3 - 2.36e3i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1−5.86e3T+2.05e7T2 1 - 5.86e3T + 2.05e7T^{2}
31 1+(−139.−241.i)T+(−1.43e7+2.47e7i)T2 1 + (-139. - 241. i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(1.51e3−2.63e3i)T+(−3.46e7−6.00e7i)T2 1 + (1.51e3 - 2.63e3i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1+819.T+1.15e8T2 1 + 819.T + 1.15e8T^{2}
43 1−1.11e4T+1.47e8T2 1 - 1.11e4T + 1.47e8T^{2}
47 1+(3.70e3−6.41e3i)T+(−1.14e8−1.98e8i)T2 1 + (3.70e3 - 6.41e3i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(−6.84e3−1.18e4i)T+(−2.09e8+3.62e8i)T2 1 + (-6.84e3 - 1.18e4i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(−1.11e4−1.93e4i)T+(−3.57e8+6.19e8i)T2 1 + (-1.11e4 - 1.93e4i)T + (-3.57e8 + 6.19e8i)T^{2}
61 1+(6.34e3−1.09e4i)T+(−4.22e8−7.31e8i)T2 1 + (6.34e3 - 1.09e4i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(−2.61e4−4.52e4i)T+(−6.75e8+1.16e9i)T2 1 + (-2.61e4 - 4.52e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1−6.02e4T+1.80e9T2 1 - 6.02e4T + 1.80e9T^{2}
73 1+(−3.84e4−6.66e4i)T+(−1.03e9+1.79e9i)T2 1 + (-3.84e4 - 6.66e4i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(−1.67e4+2.90e4i)T+(−1.53e9−2.66e9i)T2 1 + (-1.67e4 + 2.90e4i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1+6.05e4T+3.93e9T2 1 + 6.05e4T + 3.93e9T^{2}
89 1+(−4.60e4+7.98e4i)T+(−2.79e9−4.83e9i)T2 1 + (-4.60e4 + 7.98e4i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1+1.52e5T+8.58e9T2 1 + 1.52e5T + 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

−11.02229594255246187111543619416, −10.08371368363493660137410922651, −9.328099407550582134273686203077, −8.373228169046046437022322483191, −7.08979245573586115360182728680, −5.74366767397527999003523320786, −4.72409667515195608625694652029, −3.68956975617950421620194180398, −2.58413574235771668159232023638, −1.16551238210768898715563918903, 0.64467589223355718018740164248, 2.34006760858001028025383182342, 3.61278462043272959497986302497, 4.84891800199306761638731825166, 6.13166889015983905626727900124, 6.80950400419099369812627654849, 7.998391760376409433992042955477, 8.610730360258950093824357658209, 9.798378957420809606178297380592, 10.92829727864364096354099214595

Graph of the ZZ-function along the critical line