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Properties

Label 2-158-79.67-c1-0-1
Degree $2$
Conductor $158$
Sign $-0.934 - 0.355i$
Analytic cond. $1.26163$
Root an. cond. $1.12322$
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  + (0.120 + 0.992i)2-s + (−1.14 + 3.02i)3-s + (−0.970 + 0.239i)4-s + (1.75 + 0.922i)5-s + (−3.13 − 0.773i)6-s + (0.00805 − 0.0212i)7-s + (−0.354 − 0.935i)8-s + (−5.56 − 4.92i)9-s + (−0.704 + 1.85i)10-s + (2.02 − 1.06i)11-s + (0.389 − 3.20i)12-s + (3.20 − 0.789i)13-s + (0.0220 + 0.00543i)14-s + (−4.80 + 4.25i)15-s + (0.885 − 0.464i)16-s + (−3.17 + 0.782i)17-s + ⋯
L(s)  = 1  + (0.0852 + 0.701i)2-s + (−0.661 + 1.74i)3-s + (−0.485 + 0.119i)4-s + (0.786 + 0.412i)5-s + (−1.28 − 0.315i)6-s + (0.00304 − 0.00802i)7-s + (−0.125 − 0.330i)8-s + (−1.85 − 1.64i)9-s + (−0.222 + 0.587i)10-s + (0.611 − 0.320i)11-s + (0.112 − 0.925i)12-s + (0.888 − 0.219i)13-s + (0.00589 + 0.00145i)14-s + (−1.23 + 1.09i)15-s + (0.221 − 0.116i)16-s + (−0.769 + 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(158\)    =    \(2 \cdot 79\)
Sign: $-0.934 - 0.355i$
Analytic conductor: \(1.26163\)
Root analytic conductor: \(1.12322\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{158} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 158,\ (\ :1/2),\ -0.934 - 0.355i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.180743 + 0.984954i\)
\(L(\frac12)\) \(\approx\) \(0.180743 + 0.984954i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.120 - 0.992i)T \)
79 \( 1 + (3.63 + 8.10i)T \)
good3 \( 1 + (1.14 - 3.02i)T + (-2.24 - 1.98i)T^{2} \)
5 \( 1 + (-1.75 - 0.922i)T + (2.84 + 4.11i)T^{2} \)
7 \( 1 + (-0.00805 + 0.0212i)T + (-5.23 - 4.64i)T^{2} \)
11 \( 1 + (-2.02 + 1.06i)T + (6.24 - 9.05i)T^{2} \)
13 \( 1 + (-3.20 + 0.789i)T + (11.5 - 6.04i)T^{2} \)
17 \( 1 + (3.17 - 0.782i)T + (15.0 - 7.90i)T^{2} \)
19 \( 1 + (3.96 - 5.74i)T + (-6.73 - 17.7i)T^{2} \)
23 \( 1 - 5.89T + 23T^{2} \)
29 \( 1 + (-3.22 + 2.85i)T + (3.49 - 28.7i)T^{2} \)
31 \( 1 + (-1.16 - 9.62i)T + (-30.0 + 7.41i)T^{2} \)
37 \( 1 + (-5.06 + 7.33i)T + (-13.1 - 34.5i)T^{2} \)
41 \( 1 + (5.91 + 3.10i)T + (23.2 + 33.7i)T^{2} \)
43 \( 1 + (-2.41 - 1.26i)T + (24.4 + 35.3i)T^{2} \)
47 \( 1 + (2.73 + 3.96i)T + (-16.6 + 43.9i)T^{2} \)
53 \( 1 + (-0.868 - 2.28i)T + (-39.6 + 35.1i)T^{2} \)
59 \( 1 + (12.6 + 3.11i)T + (52.2 + 27.4i)T^{2} \)
61 \( 1 + (2.11 - 3.06i)T + (-21.6 - 57.0i)T^{2} \)
67 \( 1 + (-0.646 + 5.32i)T + (-65.0 - 16.0i)T^{2} \)
71 \( 1 + (3.46 + 9.14i)T + (-53.1 + 47.0i)T^{2} \)
73 \( 1 + (-15.5 - 3.83i)T + (64.6 + 33.9i)T^{2} \)
83 \( 1 + (-6.41 + 1.58i)T + (73.4 - 38.5i)T^{2} \)
89 \( 1 + (-3.68 + 9.72i)T + (-66.6 - 59.0i)T^{2} \)
97 \( 1 + (1.87 - 2.71i)T + (-34.3 - 90.6i)T^{2} \)
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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

−13.75407131980529835992738021372, −12.26997536799720673397549361591, −10.88873922648027061085783970626, −10.43754659348132564934221602590, −9.299761678452079916364256370917, −8.563361406945333014306057515305, −6.42300010204837036854149121646, −5.88528576548887871581291629000, −4.61981894472902320673830203791, −3.50949224896912171433039485675, 1.16337453741131638626628206898, 2.39798006006066825294223149545, 4.82062970662139980061279192547, 6.14593100112318850262954175397, 6.87946796306381377152388392586, 8.402442402919116783485125052077, 9.326770810885052784383831372564, 10.98918374907245353824331133173, 11.51886390471938811269103203247, 12.60658225149801601964441851519

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