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Properties

Label 2-152-8.5-c1-0-17
Degree $2$
Conductor $152$
Sign $-0.958 + 0.283i$
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  + (−0.135 − 1.40i)2-s − 1.70i·3-s + (−1.96 + 0.381i)4-s − 1.66i·5-s + (−2.40 + 0.231i)6-s − 1.99·7-s + (0.802 + 2.71i)8-s + 0.0874·9-s + (−2.33 + 0.225i)10-s + 2.08i·11-s + (0.650 + 3.35i)12-s − 4.77i·13-s + (0.270 + 2.80i)14-s − 2.83·15-s + (3.70 − 1.49i)16-s + 2.10·17-s + ⋯
L(s)  = 1  + (−0.0958 − 0.995i)2-s − 0.985i·3-s + (−0.981 + 0.190i)4-s − 0.743i·5-s + (−0.980 + 0.0943i)6-s − 0.754·7-s + (0.283 + 0.958i)8-s + 0.0291·9-s + (−0.739 + 0.0712i)10-s + 0.629i·11-s + (0.187 + 0.967i)12-s − 1.32i·13-s + (0.0722 + 0.750i)14-s − 0.732·15-s + (0.927 − 0.374i)16-s + 0.510·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.958 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.958 + 0.283i$
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: \((2,\ 152,\ (\ :1/2),\ -0.958 + 0.283i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128510 - 0.886714i\)
\(L(\frac12)\) \(\approx\) \(0.128510 - 0.886714i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.135 + 1.40i)T \)
19 \( 1 + iT \)
good3 \( 1 + 1.70iT - 3T^{2} \)
5 \( 1 + 1.66iT - 5T^{2} \)
7 \( 1 + 1.99T + 7T^{2} \)
11 \( 1 - 2.08iT - 11T^{2} \)
13 \( 1 + 4.77iT - 13T^{2} \)
17 \( 1 - 2.10T + 17T^{2} \)
23 \( 1 + 4.84T + 23T^{2} \)
29 \( 1 - 0.695iT - 29T^{2} \)
31 \( 1 - 9.77T + 31T^{2} \)
37 \( 1 - 0.0772iT - 37T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 + 1.43iT - 43T^{2} \)
47 \( 1 + 2.88T + 47T^{2} \)
53 \( 1 - 9.00iT - 53T^{2} \)
59 \( 1 - 11.5iT - 59T^{2} \)
61 \( 1 - 8.82iT - 61T^{2} \)
67 \( 1 + 1.27iT - 67T^{2} \)
71 \( 1 + 4.66T + 71T^{2} \)
73 \( 1 - 4.44T + 73T^{2} \)
79 \( 1 - 1.10T + 79T^{2} \)
83 \( 1 + 2.47iT - 83T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 + 13.9T + 97T^{2} \)
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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

−12.57314417835821991304890755675, −11.96789655968895958369669590526, −10.40971152337105292383642531328, −9.669768835662341723189480379936, −8.405148340340109822565640807436, −7.48647692514357011727311061396, −5.90026472858020964273616642257, −4.43735686766840001434717874712, −2.74923909236795286168599604780, −1.01220196122543408428068787677, 3.47366888083275812214273162905, 4.56335786290473790875656558236, 6.08134108863639513494537001072, 6.88680764295629176854684557835, 8.264388107901841495113861483100, 9.541021529183454606644273924114, 9.989626710965967396766582290239, 11.18444676812705650244048228540, 12.63775376580085887335316198186, 13.92546609693196834743634739156

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