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Properties

Label 2-152-152.107-c1-0-10
Degree $2$
Conductor $152$
Sign $0.651 - 0.758i$
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  + (0.540 + 1.30i)2-s + (2.07 − 1.19i)3-s + (−1.41 + 1.41i)4-s + (−0.418 + 0.241i)5-s + (2.68 + 2.06i)6-s + 1.56i·7-s + (−2.61 − 1.08i)8-s + (1.37 − 2.37i)9-s + (−0.541 − 0.416i)10-s + 1.24·11-s + (−1.24 + 4.62i)12-s + (1.80 − 3.11i)13-s + (−2.04 + 0.846i)14-s + (−0.579 + 1.00i)15-s + (0.0106 − 3.99i)16-s + (−1.35 − 2.34i)17-s + ⋯
L(s)  = 1  + (0.382 + 0.924i)2-s + (1.19 − 0.692i)3-s + (−0.708 + 0.706i)4-s + (−0.187 + 0.108i)5-s + (1.09 + 0.843i)6-s + 0.592i·7-s + (−0.923 − 0.384i)8-s + (0.457 − 0.793i)9-s + (−0.171 − 0.131i)10-s + 0.374·11-s + (−0.360 + 1.33i)12-s + (0.499 − 0.864i)13-s + (−0.547 + 0.226i)14-s + (−0.149 + 0.258i)15-s + (0.00265 − 0.999i)16-s + (−0.328 − 0.569i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.651 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.651 - 0.758i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ 0.651 - 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50618 + 0.692220i\)
\(L(\frac12)\) \(\approx\) \(1.50618 + 0.692220i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.540 - 1.30i)T \)
19 \( 1 + (4.25 + 0.936i)T \)
good3 \( 1 + (-2.07 + 1.19i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.418 - 0.241i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 1.56iT - 7T^{2} \)
11 \( 1 - 1.24T + 11T^{2} \)
13 \( 1 + (-1.80 + 3.11i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.35 + 2.34i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (5.52 + 3.19i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.695 - 1.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.86T + 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 + (-1.10 + 0.635i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.78 - 8.28i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.26 + 4.19i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.50 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.43 + 2.56i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.42 - 5.44i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.22 + 1.86i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.62 - 11.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.494 + 0.856i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.81 - 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + (11.1 + 6.44i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.42 + 1.40i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.14966955276849200749745675699, −12.75639216764774818943964684316, −11.39544580994971738841996888603, −9.549194602277843548684518935879, −8.574910111539846176210014603158, −7.957824334872070866341960329615, −6.88229875243301727348743790207, −5.70787292134532106616671434912, −3.98072242758900731356565373818, −2.61940441693661048170215317179, 2.13345561542539843904034886176, 3.88699785584528073193296241487, 4.17331687092325892611332105900, 6.17377398578120915523587310947, 8.054739712940247725587575474845, 8.994667272227401654940805862523, 9.808074545705887588019907936691, 10.76144809761571358426888606361, 11.79806460266898562339343354468, 13.01709691455820152399210354889

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