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Properties

Label 2-155-31.4-c1-0-0
Degree $2$
Conductor $155$
Sign $0.491 - 0.870i$
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  + (−0.766 − 2.35i)2-s + (−0.478 + 1.47i)3-s + (−3.36 + 2.44i)4-s − 5-s + 3.84·6-s + (−3.20 + 2.32i)7-s + (4.32 + 3.14i)8-s + (0.483 + 0.351i)9-s + (0.766 + 2.35i)10-s + (0.0194 − 0.0141i)11-s + (−1.99 − 6.12i)12-s + (−1.53 + 4.71i)13-s + (7.94 + 5.77i)14-s + (0.478 − 1.47i)15-s + (1.53 − 4.71i)16-s + (0.493 + 0.358i)17-s + ⋯
L(s)  = 1  + (−0.542 − 1.66i)2-s + (−0.276 + 0.851i)3-s + (−1.68 + 1.22i)4-s − 0.447·5-s + 1.56·6-s + (−1.20 + 0.879i)7-s + (1.52 + 1.11i)8-s + (0.161 + 0.117i)9-s + (0.242 + 0.746i)10-s + (0.00587 − 0.00426i)11-s + (−0.574 − 1.76i)12-s + (−0.424 + 1.30i)13-s + (2.12 + 1.54i)14-s + (0.123 − 0.380i)15-s + (0.382 − 1.17i)16-s + (0.119 + 0.0870i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.491 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $0.491 - 0.870i$
Analytic conductor: \(1.23768\)
Root analytic conductor: \(1.11251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 155,\ (\ :1/2),\ 0.491 - 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.297564 + 0.173677i\)
\(L(\frac12)\) \(\approx\) \(0.297564 + 0.173677i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
31 \( 1 + (0.899 - 5.49i)T \)
good2 \( 1 + (0.766 + 2.35i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.478 - 1.47i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (3.20 - 2.32i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-0.0194 + 0.0141i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.53 - 4.71i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.493 - 0.358i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.17 + 6.69i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (4.65 + 3.37i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.12 - 6.54i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 0.922T + 37T^{2} \)
41 \( 1 + (-1.08 - 3.32i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (1.03 + 3.18i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-2.46 + 7.58i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.97 - 5.79i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.701 - 2.15i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + 7.34T + 61T^{2} \)
67 \( 1 - 6.55T + 67T^{2} \)
71 \( 1 + (-0.670 - 0.486i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.187 + 0.135i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-12.0 - 8.73i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.68 - 8.26i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-10.8 + 7.86i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (10.7 - 7.83i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.55493048123254778279613058086, −12.02673596188684136469599484848, −10.98723778938640517099238519195, −10.20449241897719799999679942721, −9.294699783277193278680177174548, −8.747105156091704478293854784061, −6.78996570800391349421786800701, −4.82738756551864326250139775203, −3.75119972866882056236699439744, −2.43397818852550885704554166422, 0.40385531684993066242702207969, 3.93418655383420028445707952868, 5.81187193836049199286134833158, 6.48482887688793026154483311641, 7.62243979624697827937645362449, 7.912128271291711153958982234399, 9.630664919151641168819656054169, 10.25361530219166945609662915079, 12.13805987546070443545331359509, 13.02777173939362210901226175406

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