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Properties

Label 2-14e2-7.2-c5-0-11
Degree $2$
Conductor $196$
Sign $0.386 + 0.922i$
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  + (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

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

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.386 + 0.922i$
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: \((2,\ 196,\ (\ :5/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.642198392\)
\(L(\frac12)\) \(\approx\) \(2.642198392\)
\(L(\frac{7}{2})\) 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 \( 1 + (-1 - 1.73i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-48 + 83.1i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-360 - 623. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 572T + 3.71e5T^{2} \)
17 \( 1 + (627 + 1.08e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-47 + 81.4i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (48 - 83.1i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 4.37e3T + 2.05e7T^{2} \)
31 \( 1 + (-3.12e3 - 5.40e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-5.39e3 + 9.35e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 1.20e4T + 1.15e8T^{2} \)
43 \( 1 + 9.16e3T + 1.47e8T^{2} \)
47 \( 1 + (-1.29e4 + 2.23e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (507 + 878. i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (621 + 1.07e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (3.79e3 - 6.57e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (2.05e4 + 3.56e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 3.76e4T + 1.80e9T^{2} \)
73 \( 1 + (-6.71e3 - 1.16e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (3.12e3 - 5.41e3i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 2.52e4T + 3.93e9T^{2} \)
89 \( 1 + (-2.25e4 + 3.90e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 1.07e5T + 8.58e9T^{2} \)
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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

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

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