Properties

Degree $2$
Conductor $158$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s + 4-s − 2·5-s + 2·6-s + 8-s + 9-s − 2·10-s − 4·11-s + 2·12-s + 2·13-s − 4·15-s + 16-s − 2·17-s + 18-s − 2·20-s − 4·22-s + 2·24-s − 25-s + 2·26-s − 4·27-s + 8·29-s − 4·30-s + 8·31-s + 32-s − 8·33-s − 2·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s + 0.577·12-s + 0.554·13-s − 1.03·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.447·20-s − 0.852·22-s + 0.408·24-s − 1/5·25-s + 0.392·26-s − 0.769·27-s + 1.48·29-s − 0.730·30-s + 1.43·31-s + 0.176·32-s − 1.39·33-s − 0.342·34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(158\)    =    \(2 \cdot 79\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{158} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 158,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.916415273\)
\(L(\frac12)\) \(\approx\) \(1.916415273\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
79 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−19.46485786324276, −18.74429396831904, −17.54434164537235, −16.02747471441232, −15.59217826695895, −14.81413470540775, −13.76505972948267, −13.22100070384617, −12.04877850935267, −11.09280087830307, −9.913734664908443, −8.342366728750083, −7.992164212881315, −6.573069489630046, −4.924485662312492, −3.652973568677172, −2.587672816813015, 2.587672816813015, 3.652973568677172, 4.924485662312492, 6.573069489630046, 7.992164212881315, 8.342366728750083, 9.913734664908443, 11.09280087830307, 12.04877850935267, 13.22100070384617, 13.76505972948267, 14.81413470540775, 15.59217826695895, 16.02747471441232, 17.54434164537235, 18.74429396831904, 19.46485786324276

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