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Properties

Label 2-14e2-7.4-c5-0-7
Degree 22
Conductor 196196
Sign 0.386−0.922i0.386 - 0.922i
Analytic cond. 31.435231.4352
Root an. cond. 5.606715.60671
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  + (1 − 1.73i)3-s + (48 + 83.1i)5-s + (119.5 + 206. i)9-s + (360 − 623. i)11-s + 572·13-s + 192·15-s + (−627 + 1.08e3i)17-s + (47 + 81.4i)19-s + (−48 − 83.1i)23-s + (−3.04e3 + 5.27e3i)25-s + 964·27-s − 4.37e3·29-s + (3.12e3 − 5.40e3i)31-s + (−720 − 1.24e3i)33-s + (5.39e3 + 9.35e3i)37-s + ⋯
L(s)  = 1  + (0.0641 − 0.111i)3-s + (0.858 + 1.48i)5-s + (0.491 + 0.851i)9-s + (0.897 − 1.55i)11-s + 0.938·13-s + 0.220·15-s + (−0.526 + 0.911i)17-s + (0.0298 + 0.0517i)19-s + (−0.0189 − 0.0327i)23-s + (−0.974 + 1.68i)25-s + 0.254·27-s − 0.965·29-s + (0.583 − 1.01i)31-s + (−0.115 − 0.199i)33-s + (0.648 + 1.12i)37-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((0.386−0.922i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+5/2) L(s)=((0.386−0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: 0.386−0.922i0.386 - 0.922i
Analytic conductor: 31.435231.4352
Root analytic conductor: 5.606715.60671
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ196(165,⋅)\chi_{196} (165, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 196, ( :5/2), 0.386−0.922i)(2,\ 196,\ (\ :5/2),\ 0.386 - 0.922i)

Particular Values

L(3)L(3) ≈\approx 2.6421983922.642198392
L(12)L(\frac12) ≈\approx 2.6421983922.642198392
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 1
7 1 1
good3 1+(−1+1.73i)T+(−121.5−210.i)T2 1 + (-1 + 1.73i)T + (-121.5 - 210. i)T^{2}
5 1+(−48−83.1i)T+(−1.56e3+2.70e3i)T2 1 + (-48 - 83.1i)T + (-1.56e3 + 2.70e3i)T^{2}
11 1+(−360+623.i)T+(−8.05e4−1.39e5i)T2 1 + (-360 + 623. i)T + (-8.05e4 - 1.39e5i)T^{2}
13 1−572T+3.71e5T2 1 - 572T + 3.71e5T^{2}
17 1+(627−1.08e3i)T+(−7.09e5−1.22e6i)T2 1 + (627 - 1.08e3i)T + (-7.09e5 - 1.22e6i)T^{2}
19 1+(−47−81.4i)T+(−1.23e6+2.14e6i)T2 1 + (-47 - 81.4i)T + (-1.23e6 + 2.14e6i)T^{2}
23 1+(48+83.1i)T+(−3.21e6+5.57e6i)T2 1 + (48 + 83.1i)T + (-3.21e6 + 5.57e6i)T^{2}
29 1+4.37e3T+2.05e7T2 1 + 4.37e3T + 2.05e7T^{2}
31 1+(−3.12e3+5.40e3i)T+(−1.43e7−2.47e7i)T2 1 + (-3.12e3 + 5.40e3i)T + (-1.43e7 - 2.47e7i)T^{2}
37 1+(−5.39e3−9.35e3i)T+(−3.46e7+6.00e7i)T2 1 + (-5.39e3 - 9.35e3i)T + (-3.46e7 + 6.00e7i)T^{2}
41 1−1.20e4T+1.15e8T2 1 - 1.20e4T + 1.15e8T^{2}
43 1+9.16e3T+1.47e8T2 1 + 9.16e3T + 1.47e8T^{2}
47 1+(−1.29e4−2.23e4i)T+(−1.14e8+1.98e8i)T2 1 + (-1.29e4 - 2.23e4i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+(507−878.i)T+(−2.09e8−3.62e8i)T2 1 + (507 - 878. i)T + (-2.09e8 - 3.62e8i)T^{2}
59 1+(621−1.07e3i)T+(−3.57e8−6.19e8i)T2 1 + (621 - 1.07e3i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(3.79e3+6.57e3i)T+(−4.22e8+7.31e8i)T2 1 + (3.79e3 + 6.57e3i)T + (-4.22e8 + 7.31e8i)T^{2}
67 1+(2.05e4−3.56e4i)T+(−6.75e8−1.16e9i)T2 1 + (2.05e4 - 3.56e4i)T + (-6.75e8 - 1.16e9i)T^{2}
71 1+3.76e4T+1.80e9T2 1 + 3.76e4T + 1.80e9T^{2}
73 1+(−6.71e3+1.16e4i)T+(−1.03e9−1.79e9i)T2 1 + (-6.71e3 + 1.16e4i)T + (-1.03e9 - 1.79e9i)T^{2}
79 1+(3.12e3+5.41e3i)T+(−1.53e9+2.66e9i)T2 1 + (3.12e3 + 5.41e3i)T + (-1.53e9 + 2.66e9i)T^{2}
83 1+2.52e4T+3.93e9T2 1 + 2.52e4T + 3.93e9T^{2}
89 1+(−2.25e4−3.90e4i)T+(−2.79e9+4.83e9i)T2 1 + (-2.25e4 - 3.90e4i)T + (-2.79e9 + 4.83e9i)T^{2}
97 1−1.07e5T+8.58e9T2 1 - 1.07e5T + 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

−11.32464509502692729273788668847, −10.94867083935413610935020352043, −10.00834919721597987986457834928, −8.816814517174552640553453559451, −7.66335222581771956890716831223, −6.34816407409272334377585672761, −5.96517132056709824531322519630, −3.96232046359291411778283530961, −2.76775958912000936925935696216, −1.45585746379722354554727224778, 0.895393808723037845062778731143, 1.89389763388040805378306134700, 4.00948766188503505360097222881, 4.88413329179933057710171367319, 6.14327969513421458966312187946, 7.23787286664908900574329845401, 8.903823049369117879779130538055, 9.246517589269486315574353788281, 10.11822321942125499940221524956, 11.69892028731118300029891720455

Graph of the ZZ-function along the critical line