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Properties

Label 2-152-152.107-c1-0-3
Degree $2$
Conductor $152$
Sign $-0.977 - 0.211i$
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  + (0.707 + 1.22i)2-s + (−1.5 + 0.866i)3-s + (−0.999 + 1.73i)4-s + (−2.12 + 1.22i)5-s + (−2.12 − 1.22i)6-s − 2.44i·7-s − 2.82·8-s + (−3 − 1.73i)10-s + 5·11-s − 3.46i·12-s + (−2.82 + 4.89i)13-s + (2.99 − 1.73i)14-s + (2.12 − 3.67i)15-s + (−2.00 − 3.46i)16-s + (2 + 3.46i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)2-s + (−0.866 + 0.499i)3-s + (−0.499 + 0.866i)4-s + (−0.948 + 0.547i)5-s + (−0.866 − 0.499i)6-s − 0.925i·7-s − 0.999·8-s + (−0.948 − 0.547i)10-s + 1.50·11-s − 0.999i·12-s + (−0.784 + 1.35i)13-s + (0.801 − 0.462i)14-s + (0.547 − 0.948i)15-s + (−0.500 − 0.866i)16-s + (0.485 + 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.977 - 0.211i$
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: \((2,\ 152,\ (\ :1/2),\ -0.977 - 0.211i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0834372 + 0.781026i\)
\(L(\frac12)\) \(\approx\) \(0.0834372 + 0.781026i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 1.22i)T \)
19 \( 1 + (0.5 - 4.33i)T \)
good3 \( 1 + (1.5 - 0.866i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.12 - 1.22i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 2.44iT - 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 + (2.82 - 4.89i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.12 - 1.22i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.53 + 6.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.41T + 31T^{2} \)
37 \( 1 - 1.41T + 37T^{2} \)
41 \( 1 + (-4.5 + 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3 + 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.36 - 3.67i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.24 - 7.34i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 - 0.866i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.36 - 3.67i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.5 + 6.06i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.41 + 2.44i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.5 + 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.41 - 2.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7T + 83T^{2} \)
89 \( 1 + (-3 - 1.73i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (16.5 - 9.52i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.84220901977677266262687628918, −12.13476170631604144779564659743, −11.73481367067954266388151754810, −10.61813537195090453551656097049, −9.333581547176024820741410679202, −7.85437630839799527345455859732, −6.96341717718549937853566256241, −6.00407473949427512806681082332, −4.36176876124625463964234614868, −3.88356897410588603042316829822, 0.77045438857431999662270436027, 3.07864682760469755749302856356, 4.70913580745705735068031008599, 5.65564817872870067851478131956, 6.93093610725498866336159873097, 8.602464246529525189107673530702, 9.515772897412458442440891349222, 11.03270780934150752150794651758, 11.86790695529451717153412319223, 12.22321603335649059991797184286

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