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Properties

Label 2-68-17.8-c5-0-1
Degree $2$
Conductor $68$
Sign $0.478 - 0.878i$
Analytic cond. $10.9060$
Root an. cond. $3.30243$
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  + (−1.40 + 3.39i)3-s + (−69.4 − 28.7i)5-s + (120. − 49.9i)7-s + (162. + 162. i)9-s + (148. + 359. i)11-s + 540. i·13-s + (195. − 195. i)15-s + (−647. + 1.00e3i)17-s + (790. − 790. i)19-s + 479. i·21-s + (1.45e3 + 3.51e3i)23-s + (1.78e3 + 1.78e3i)25-s + (−1.60e3 + 664. i)27-s + (6.10e3 + 2.52e3i)29-s + (426. − 1.02e3i)31-s + ⋯
L(s)  = 1  + (−0.0901 + 0.217i)3-s + (−1.24 − 0.514i)5-s + (0.930 − 0.385i)7-s + (0.667 + 0.667i)9-s + (0.371 + 0.895i)11-s + 0.887i·13-s + (0.224 − 0.224i)15-s + (−0.543 + 0.839i)17-s + (0.502 − 0.502i)19-s + 0.237i·21-s + (0.573 + 1.38i)23-s + (0.571 + 0.571i)25-s + (−0.423 + 0.175i)27-s + (1.34 + 0.558i)29-s + (0.0796 − 0.192i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.478 - 0.878i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $0.478 - 0.878i$
Analytic conductor: \(10.9060\)
Root analytic conductor: \(3.30243\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{68} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 68,\ (\ :5/2),\ 0.478 - 0.878i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.22345 + 0.726807i\)
\(L(\frac12)\) \(\approx\) \(1.22345 + 0.726807i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (647. - 1.00e3i)T \)
good3 \( 1 + (1.40 - 3.39i)T + (-171. - 171. i)T^{2} \)
5 \( 1 + (69.4 + 28.7i)T + (2.20e3 + 2.20e3i)T^{2} \)
7 \( 1 + (-120. + 49.9i)T + (1.18e4 - 1.18e4i)T^{2} \)
11 \( 1 + (-148. - 359. i)T + (-1.13e5 + 1.13e5i)T^{2} \)
13 \( 1 - 540. iT - 3.71e5T^{2} \)
19 \( 1 + (-790. + 790. i)T - 2.47e6iT^{2} \)
23 \( 1 + (-1.45e3 - 3.51e3i)T + (-4.55e6 + 4.55e6i)T^{2} \)
29 \( 1 + (-6.10e3 - 2.52e3i)T + (1.45e7 + 1.45e7i)T^{2} \)
31 \( 1 + (-426. + 1.02e3i)T + (-2.02e7 - 2.02e7i)T^{2} \)
37 \( 1 + (-4.91e3 + 1.18e4i)T + (-4.90e7 - 4.90e7i)T^{2} \)
41 \( 1 + (1.62e3 - 673. i)T + (8.19e7 - 8.19e7i)T^{2} \)
43 \( 1 + (5.41e3 + 5.41e3i)T + 1.47e8iT^{2} \)
47 \( 1 - 2.91e4iT - 2.29e8T^{2} \)
53 \( 1 + (-1.75e4 + 1.75e4i)T - 4.18e8iT^{2} \)
59 \( 1 + (2.54e4 + 2.54e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (3.66e4 - 1.51e4i)T + (5.97e8 - 5.97e8i)T^{2} \)
67 \( 1 + 3.64e4T + 1.35e9T^{2} \)
71 \( 1 + (4.51e3 - 1.08e4i)T + (-1.27e9 - 1.27e9i)T^{2} \)
73 \( 1 + (-5.78e4 - 2.39e4i)T + (1.46e9 + 1.46e9i)T^{2} \)
79 \( 1 + (-2.12e4 - 5.12e4i)T + (-2.17e9 + 2.17e9i)T^{2} \)
83 \( 1 + (1.72e4 - 1.72e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 3.34e4iT - 5.58e9T^{2} \)
97 \( 1 + (1.99e3 + 824. i)T + (6.07e9 + 6.07e9i)T^{2} \)
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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

−14.03639077134539053770356575167, −12.73447808838408842217679882744, −11.65353396028463772563092075736, −10.83996970180961026147443125908, −9.310110806075493229923291101685, −7.973800701623393630317503420206, −7.12185270143155671166281663240, −4.79510435773825861684075325976, −4.13841228340863525281169016827, −1.49040505901631689996703218900, 0.74759684695704377296967772344, 3.13965485650273850859188995669, 4.64451355947134946440694065369, 6.46931050018694895548976825567, 7.71976069850780592621359695109, 8.671079822432321806074910686878, 10.42448715139487505104819888912, 11.58954468793929473244264718793, 12.12956464133445650572420356102, 13.65927762917862769412318519420

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