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Properties

Label 2-158-79.8-c1-0-1
Degree $2$
Conductor $158$
Sign $0.270 - 0.962i$
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  + (−0.354 + 0.935i)2-s + (2.70 + 1.42i)3-s + (−0.748 − 0.663i)4-s + (0.0216 − 0.177i)5-s + (−2.28 + 2.02i)6-s + (−0.585 − 0.307i)7-s + (0.885 − 0.464i)8-s + (3.60 + 5.22i)9-s + (0.158 + 0.0833i)10-s + (−0.199 − 1.64i)11-s + (−1.08 − 2.85i)12-s + (−4.38 − 3.88i)13-s + (0.494 − 0.438i)14-s + (0.311 − 0.451i)15-s + (0.120 + 0.992i)16-s + (1.30 + 1.15i)17-s + ⋯
L(s)  = 1  + (−0.250 + 0.661i)2-s + (1.56 + 0.820i)3-s + (−0.374 − 0.331i)4-s + (0.00966 − 0.0795i)5-s + (−0.934 + 0.827i)6-s + (−0.221 − 0.116i)7-s + (0.313 − 0.164i)8-s + (1.20 + 1.74i)9-s + (0.0501 + 0.0263i)10-s + (−0.0601 − 0.495i)11-s + (−0.312 − 0.825i)12-s + (−1.21 − 1.07i)13-s + (0.132 − 0.117i)14-s + (0.0803 − 0.116i)15-s + (0.0301 + 0.248i)16-s + (0.316 + 0.280i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(158\)    =    \(2 \cdot 79\)
Sign: $0.270 - 0.962i$
Analytic conductor: \(1.26163\)
Root analytic conductor: \(1.12322\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{158} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 158,\ (\ :1/2),\ 0.270 - 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15355 + 0.874090i\)
\(L(\frac12)\) \(\approx\) \(1.15355 + 0.874090i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.354 - 0.935i)T \)
79 \( 1 + (0.848 - 8.84i)T \)
good3 \( 1 + (-2.70 - 1.42i)T + (1.70 + 2.46i)T^{2} \)
5 \( 1 + (-0.0216 + 0.177i)T + (-4.85 - 1.19i)T^{2} \)
7 \( 1 + (0.585 + 0.307i)T + (3.97 + 5.76i)T^{2} \)
11 \( 1 + (0.199 + 1.64i)T + (-10.6 + 2.63i)T^{2} \)
13 \( 1 + (4.38 + 3.88i)T + (1.56 + 12.9i)T^{2} \)
17 \( 1 + (-1.30 - 1.15i)T + (2.04 + 16.8i)T^{2} \)
19 \( 1 + (3.81 - 0.940i)T + (16.8 - 8.82i)T^{2} \)
23 \( 1 + 1.98T + 23T^{2} \)
29 \( 1 + (-4.72 + 6.84i)T + (-10.2 - 27.1i)T^{2} \)
31 \( 1 + (1.45 - 3.83i)T + (-23.2 - 20.5i)T^{2} \)
37 \( 1 + (8.02 - 1.97i)T + (32.7 - 17.1i)T^{2} \)
41 \( 1 + (0.377 - 3.10i)T + (-39.8 - 9.81i)T^{2} \)
43 \( 1 + (-1.05 + 8.72i)T + (-41.7 - 10.2i)T^{2} \)
47 \( 1 + (-12.0 - 2.96i)T + (41.6 + 21.8i)T^{2} \)
53 \( 1 + (-8.58 + 4.50i)T + (30.1 - 43.6i)T^{2} \)
59 \( 1 + (0.592 - 0.524i)T + (7.11 - 58.5i)T^{2} \)
61 \( 1 + (7.04 - 1.73i)T + (54.0 - 28.3i)T^{2} \)
67 \( 1 + (-5.34 - 14.0i)T + (-50.1 + 44.4i)T^{2} \)
71 \( 1 + (11.2 - 5.92i)T + (40.3 - 58.4i)T^{2} \)
73 \( 1 + (2.66 - 2.35i)T + (8.79 - 72.4i)T^{2} \)
83 \( 1 + (-9.55 - 8.46i)T + (10.0 + 82.3i)T^{2} \)
89 \( 1 + (14.7 + 7.72i)T + (50.5 + 73.2i)T^{2} \)
97 \( 1 + (9.55 - 2.35i)T + (85.8 - 45.0i)T^{2} \)
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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

−13.50780009828121501915162574022, −12.45041652896368913530914477558, −10.40643390507088164975925919595, −10.02735538245476528927762765741, −8.775294709015074447566783502524, −8.217119625988190283567966030225, −7.13227920974614728007594339534, −5.37118518229445498451910779653, −4.05710630096383017071240868049, −2.71382925172776535146506231874, 1.94482000786662712034936209443, 2.94916300656304481448812884417, 4.44596220493018769930407727000, 6.81167328910985396520363285207, 7.58749759753512964112808644624, 8.810594387136494009395678315242, 9.367597490647262135355363432690, 10.50328242864984639346764604924, 12.22296071483646899837531694284, 12.50498345048678491312254456751

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