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Properties

Label 2-152-152.107-c1-0-4
Degree $2$
Conductor $152$
Sign $-0.124 - 0.992i$
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  + (1.41 − 0.0756i)2-s + (−2.65 + 1.53i)3-s + (1.98 − 0.213i)4-s + (−2.25 + 1.30i)5-s + (−3.63 + 2.36i)6-s + 4.30i·7-s + (2.79 − 0.452i)8-s + (3.20 − 5.54i)9-s + (−3.08 + 2.01i)10-s − 0.349·11-s + (−4.95 + 3.61i)12-s + (0.839 − 1.45i)13-s + (0.325 + 6.07i)14-s + (3.99 − 6.92i)15-s + (3.90 − 0.850i)16-s + (0.357 + 0.618i)17-s + ⋯
L(s)  = 1  + (0.998 − 0.0535i)2-s + (−1.53 + 0.885i)3-s + (0.994 − 0.106i)4-s + (−1.00 + 0.582i)5-s + (−1.48 + 0.965i)6-s + 1.62i·7-s + (0.987 − 0.159i)8-s + (1.06 − 1.84i)9-s + (−0.976 + 0.635i)10-s − 0.105·11-s + (−1.42 + 1.04i)12-s + (0.232 − 0.403i)13-s + (0.0870 + 1.62i)14-s + (1.03 − 1.78i)15-s + (0.977 − 0.212i)16-s + (0.0865 + 0.149i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.124 - 0.992i$
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: \((2,\ 152,\ (\ :1/2),\ -0.124 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.728185 + 0.824858i\)
\(L(\frac12)\) \(\approx\) \(0.728185 + 0.824858i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0756i)T \)
19 \( 1 + (-4.35 + 0.268i)T \)
good3 \( 1 + (2.65 - 1.53i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.25 - 1.30i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 4.30iT - 7T^{2} \)
11 \( 1 + 0.349T + 11T^{2} \)
13 \( 1 + (-0.839 + 1.45i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.357 - 0.618i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.38 + 0.797i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.463 + 0.803i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.80T + 31T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 + (-4.87 + 2.81i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.32 - 7.48i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.26 + 3.61i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.73 - 8.20i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.62 + 3.82i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.46 + 2.00i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.6 + 6.12i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.09 + 8.82i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.54 - 4.40i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.31 + 2.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.69T + 83T^{2} \)
89 \( 1 + (-7.37 - 4.25i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.30 + 3.63i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.80542095474111825965166752800, −11.99641092731616482602312294723, −11.46242512035260745658063168735, −10.79925316258895977062719451524, −9.544399062332076676955606714897, −7.70352351658690313531356906566, −6.23225247455229623948171538106, −5.58078426714627991885852290604, −4.48518694295945119368423262262, −3.15395029256111904059506390777, 1.04175276903995945481927236818, 3.98166891084998513451631797396, 4.88996538710843593277091480487, 6.16537333575942203225387311815, 7.30305998676960456153843613085, 7.71160625671602733460123175240, 10.27772712848495111290188076587, 11.32367061408005558402984489455, 11.68058095069953966327887904194, 12.74852636901948811135346104746

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