Properties

Label 2-155-155.37-c1-0-13
Degree $2$
Conductor $155$
Sign $-0.913 - 0.406i$
Analytic cond. $1.23768$
Root an. cond. $1.11251$
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  + (−1.51 − 1.51i)2-s + (0.852 − 3.18i)3-s + 2.59i·4-s + (−1.57 + 1.58i)5-s + (−6.11 + 3.53i)6-s + (0.903 − 3.37i)7-s + (0.909 − 0.909i)8-s + (−6.78 − 3.91i)9-s + (4.79 − 0.00681i)10-s + (2.73 + 1.58i)11-s + (8.26 + 2.21i)12-s + (−2.14 + 0.573i)13-s + (−6.48 + 3.74i)14-s + (3.68 + 6.37i)15-s + 2.44·16-s + (2.29 + 0.615i)17-s + ⋯
L(s)  = 1  + (−1.07 − 1.07i)2-s + (0.491 − 1.83i)3-s + 1.29i·4-s + (−0.706 + 0.708i)5-s + (−2.49 + 1.44i)6-s + (0.341 − 1.27i)7-s + (0.321 − 0.321i)8-s + (−2.26 − 1.30i)9-s + (1.51 − 0.00215i)10-s + (0.825 + 0.476i)11-s + (2.38 + 0.639i)12-s + (−0.594 + 0.159i)13-s + (−1.73 + 0.999i)14-s + (0.952 + 1.64i)15-s + 0.610·16-s + (0.557 + 0.149i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $-0.913 - 0.406i$
Analytic conductor: \(1.23768\)
Root analytic conductor: \(1.11251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 155,\ (\ :1/2),\ -0.913 - 0.406i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.130521 + 0.614000i\)
\(L(\frac12)\) \(\approx\) \(0.130521 + 0.614000i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.57 - 1.58i)T \)
31 \( 1 + (-2.33 + 5.05i)T \)
good2 \( 1 + (1.51 + 1.51i)T + 2iT^{2} \)
3 \( 1 + (-0.852 + 3.18i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-0.903 + 3.37i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-2.73 - 1.58i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.14 - 0.573i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-2.29 - 0.615i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.05 + 1.18i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.57 + 1.57i)T + 23iT^{2} \)
29 \( 1 + 2.26T + 29T^{2} \)
37 \( 1 + (-3.52 - 0.943i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.995 + 1.72i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.494 + 1.84i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (2.97 + 2.97i)T + 47iT^{2} \)
53 \( 1 + (1.25 - 0.336i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.22 + 0.708i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 5.71iT - 61T^{2} \)
67 \( 1 + (-7.80 + 2.09i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.08 + 1.88i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.40 - 0.913i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.13 - 3.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-16.7 + 4.49i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 10.1T + 89T^{2} \)
97 \( 1 + (-8.87 - 8.87i)T + 97iT^{2} \)
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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

−11.96530639082390507528585382125, −11.63999138583194923454298768142, −10.48899258820904233838438485727, −9.285344543471263385368639377757, −7.943491266116722822966285407106, −7.55091671764308142796767120412, −6.58128740515189894084601418305, −3.64886643862653503804085652262, −2.26494056089094817458030014856, −0.854499234937580292002424850659, 3.41295228035458571050565790011, 4.91868742947584353982469053515, 5.80684271651218663952304308301, 7.84354180779333602812145704206, 8.560401519792843286412810608151, 9.234048774537627516698439312249, 9.843677287571665397584968490727, 11.27072399533020302251244834417, 12.19160909897523531106437436567, 14.28819348701847689422080040035

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