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Properties

Label 2-152-8.5-c1-0-15
Degree $2$
Conductor $152$
Sign $0.892 + 0.451i$
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  + (1.09 + 0.889i)2-s − 2.32i·3-s + (0.418 + 1.95i)4-s − 3.13i·5-s + (2.07 − 2.56i)6-s − 0.535·7-s + (−1.27 + 2.52i)8-s − 2.42·9-s + (2.79 − 3.45i)10-s + 0.425i·11-s + (4.55 − 0.975i)12-s + 6.65i·13-s + (−0.589 − 0.476i)14-s − 7.31·15-s + (−3.64 + 1.63i)16-s + 7.33·17-s + ⋯
L(s)  = 1  + (0.777 + 0.628i)2-s − 1.34i·3-s + (0.209 + 0.977i)4-s − 1.40i·5-s + (0.845 − 1.04i)6-s − 0.202·7-s + (−0.451 + 0.892i)8-s − 0.808·9-s + (0.882 − 1.09i)10-s + 0.128i·11-s + (1.31 − 0.281i)12-s + 1.84i·13-s + (−0.157 − 0.127i)14-s − 1.88·15-s + (−0.912 + 0.409i)16-s + 1.77·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.892 + 0.451i$
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: \((2,\ 152,\ (\ :1/2),\ 0.892 + 0.451i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56268 - 0.373298i\)
\(L(\frac12)\) \(\approx\) \(1.56268 - 0.373298i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.09 - 0.889i)T \)
19 \( 1 - iT \)
good3 \( 1 + 2.32iT - 3T^{2} \)
5 \( 1 + 3.13iT - 5T^{2} \)
7 \( 1 + 0.535T + 7T^{2} \)
11 \( 1 - 0.425iT - 11T^{2} \)
13 \( 1 - 6.65iT - 13T^{2} \)
17 \( 1 - 7.33T + 17T^{2} \)
23 \( 1 + 5.90T + 23T^{2} \)
29 \( 1 - 0.837iT - 29T^{2} \)
31 \( 1 + 3.16T + 31T^{2} \)
37 \( 1 + 3.49iT - 37T^{2} \)
41 \( 1 + 0.123T + 41T^{2} \)
43 \( 1 + 5.39iT - 43T^{2} \)
47 \( 1 + 2.02T + 47T^{2} \)
53 \( 1 - 5.82iT - 53T^{2} \)
59 \( 1 + 5.56iT - 59T^{2} \)
61 \( 1 + 6.99iT - 61T^{2} \)
67 \( 1 - 12.3iT - 67T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
73 \( 1 + 6.99T + 73T^{2} \)
79 \( 1 - 2.07T + 79T^{2} \)
83 \( 1 - 11.8iT - 83T^{2} \)
89 \( 1 - 13.7T + 89T^{2} \)
97 \( 1 - 0.801T + 97T^{2} \)
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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

−12.82288174002903814324707004355, −12.28531874559828336575279923249, −11.71479304650164641438357063042, −9.481657296868717001534053550526, −8.358780889100030149303973757183, −7.52433036905973346575589863793, −6.45766753728402549368000361772, −5.37437476092983389860247030998, −4.02133115179029488749776409348, −1.77424552184790539447964079363, 3.02581813848704798331632236622, 3.60142182447235088395660189524, 5.19798818469485790370357639365, 6.15545298036794654929047130583, 7.75880747481431095415227548801, 9.711162579858390525191769012331, 10.26142514170165694289134071556, 10.79009708575861360061766901886, 11.89279436518596007918744037654, 13.08467998224796132644814198585

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