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Properties

Label 2-152-8.5-c1-0-12
Degree $2$
Conductor $152$
Sign $-0.666 + 0.745i$
Analytic cond. $1.21372$
Root an. cond. $1.10169$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.37 − 0.340i)2-s − 2.95i·3-s + (1.76 + 0.935i)4-s − 2.13i·5-s + (−1.00 + 4.05i)6-s + 3.29·7-s + (−2.10 − 1.88i)8-s − 5.71·9-s + (−0.727 + 2.93i)10-s + 3.71i·11-s + (2.75 − 5.21i)12-s − 2.32i·13-s + (−4.52 − 1.12i)14-s − 6.30·15-s + (2.25 + 3.30i)16-s − 6.48·17-s + ⋯
L(s)  = 1  + (−0.970 − 0.240i)2-s − 1.70i·3-s + (0.883 + 0.467i)4-s − 0.954i·5-s + (−0.410 + 1.65i)6-s + 1.24·7-s + (−0.745 − 0.666i)8-s − 1.90·9-s + (−0.229 + 0.926i)10-s + 1.11i·11-s + (0.796 − 1.50i)12-s − 0.645i·13-s + (−1.20 − 0.299i)14-s − 1.62·15-s + (0.562 + 0.826i)16-s − 1.57·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.666 + 0.745i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ -0.666 + 0.745i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.319865 - 0.715223i\)
\(L(\frac12)\) \(\approx\) \(0.319865 - 0.715223i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 + 0.340i)T \)
19 \( 1 - iT \)
good3 \( 1 + 2.95iT - 3T^{2} \)
5 \( 1 + 2.13iT - 5T^{2} \)
7 \( 1 - 3.29T + 7T^{2} \)
11 \( 1 - 3.71iT - 11T^{2} \)
13 \( 1 + 2.32iT - 13T^{2} \)
17 \( 1 + 6.48T + 17T^{2} \)
23 \( 1 - 7.32T + 23T^{2} \)
29 \( 1 - 2.59iT - 29T^{2} \)
31 \( 1 + 1.34T + 31T^{2} \)
37 \( 1 - 3.72iT - 37T^{2} \)
41 \( 1 - 6.52T + 41T^{2} \)
43 \( 1 - 1.97iT - 43T^{2} \)
47 \( 1 - 5.45T + 47T^{2} \)
53 \( 1 + 4.98iT - 53T^{2} \)
59 \( 1 + 9.67iT - 59T^{2} \)
61 \( 1 + 8.15iT - 61T^{2} \)
67 \( 1 - 0.524iT - 67T^{2} \)
71 \( 1 - 7.17T + 71T^{2} \)
73 \( 1 + 6.33T + 73T^{2} \)
79 \( 1 + 8.75T + 79T^{2} \)
83 \( 1 - 7.74iT - 83T^{2} \)
89 \( 1 + 1.04T + 89T^{2} \)
97 \( 1 + 0.117T + 97T^{2} \)
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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

−12.72363984479850531788150847157, −11.66517573519543105559227542927, −10.86884156276505503883212450782, −9.129315130546640986797460690220, −8.348039682565060900349992146676, −7.55134339678654027225368003027, −6.65818131484262680040020965325, −4.97919659711942995010298100627, −2.25104613418163899567233956431, −1.16657140684239661570957978392, 2.75629307518976522902126641723, 4.43485219417660862855561285787, 5.75996905797368841582008312275, 7.14301192515483624549686737267, 8.663485546271885389696730488462, 9.108112932187324377533644575775, 10.50310187869545415788050583830, 11.11390272329145661545143480805, 11.35067875003436018431033243982, 13.91686206128075850927640301389

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