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Properties

Label 2-14e2-7.2-c5-0-12
Degree $2$
Conductor $196$
Sign $0.991 + 0.126i$
Analytic cond. $31.4352$
Root an. cond. $5.60671$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (12.2 + 21.1i)3-s + (41.8 − 72.4i)5-s + (−177. + 306. i)9-s + (−274. − 475. i)11-s + 823.·13-s + 2.04e3·15-s + (72.7 + 126. i)17-s + (884. − 1.53e3i)19-s + (761. − 1.31e3i)23-s + (−1.93e3 − 3.35e3i)25-s − 2.72e3·27-s − 741.·29-s + (−1.60e3 − 2.77e3i)31-s + (6.71e3 − 1.16e4i)33-s + (1.77e3 − 3.07e3i)37-s + ⋯
L(s)  = 1  + (0.783 + 1.35i)3-s + (0.748 − 1.29i)5-s + (−0.729 + 1.26i)9-s + (−0.684 − 1.18i)11-s + 1.35·13-s + 2.34·15-s + (0.0610 + 0.105i)17-s + (0.561 − 0.973i)19-s + (0.300 − 0.519i)23-s + (−0.619 − 1.07i)25-s − 0.718·27-s − 0.163·29-s + (−0.299 − 0.518i)31-s + (1.07 − 1.85i)33-s + (0.213 − 0.369i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(31.4352\)
Root analytic conductor: \(5.60671\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :5/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.033935160\)
\(L(\frac12)\) \(\approx\) \(3.033935160\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-12.2 - 21.1i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-41.8 + 72.4i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (274. + 475. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 823.T + 3.71e5T^{2} \)
17 \( 1 + (-72.7 - 126. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-884. + 1.53e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-761. + 1.31e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 741.T + 2.05e7T^{2} \)
31 \( 1 + (1.60e3 + 2.77e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-1.77e3 + 3.07e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 6.46e3T + 1.15e8T^{2} \)
43 \( 1 - 6.71e3T + 1.47e8T^{2} \)
47 \( 1 + (9.58e3 - 1.66e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.06e4 - 1.84e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.82e4 - 3.16e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-2.17e4 + 3.76e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (2.51e3 + 4.35e3i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 6.31e3T + 1.80e9T^{2} \)
73 \( 1 + (2.12e4 + 3.68e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (4.75e4 - 8.23e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 3.49e4T + 3.93e9T^{2} \)
89 \( 1 + (-5.02e4 + 8.70e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 7.67e4T + 8.58e9T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.26249102701410665318967534215, −10.49145600938494469551324899136, −9.300354197523794710945454286737, −8.906896134595569049464746759102, −8.077425330578702060762731979321, −5.94093316598975188013158709702, −5.07872375980660560436827936180, −3.98624790416189451927023700217, −2.75856060302856367688561562366, −0.918847222084727956440499989217, 1.45734798593070834679180916297, 2.37486030542028861721517428682, 3.45095474373236697540675101419, 5.67966837926658017489639476957, 6.74441911318821707790377075646, 7.38743844620627106113580737982, 8.372967261820842136163952006827, 9.686545878591739211908999717339, 10.58144857588102367692084709171, 11.75835028836693752126735452666

Graph of the $Z$-function along the critical line