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Properties

Label 2-154-11.9-c1-0-1
Degree $2$
Conductor $154$
Sign $0.868 - 0.494i$
Analytic cond. $1.22969$
Root an. cond. $1.10891$
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  + (−0.309 + 0.951i)2-s + (1.89 − 1.38i)3-s + (−0.809 − 0.587i)4-s + (0.865 + 2.66i)5-s + (0.725 + 2.23i)6-s + (−0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (0.777 − 2.39i)9-s − 2.79·10-s + (1.50 + 2.95i)11-s − 2.34·12-s + (2.18 − 6.72i)13-s + (0.809 − 0.587i)14-s + (5.31 + 3.86i)15-s + (0.309 + 0.951i)16-s + (−0.850 − 2.61i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (1.09 − 0.796i)3-s + (−0.404 − 0.293i)4-s + (0.386 + 1.19i)5-s + (0.296 + 0.911i)6-s + (−0.305 − 0.222i)7-s + (0.286 − 0.207i)8-s + (0.259 − 0.797i)9-s − 0.885·10-s + (0.454 + 0.890i)11-s − 0.677·12-s + (0.605 − 1.86i)13-s + (0.216 − 0.157i)14-s + (1.37 + 0.997i)15-s + (0.0772 + 0.237i)16-s + (−0.206 − 0.634i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $0.868 - 0.494i$
Analytic conductor: \(1.22969\)
Root analytic conductor: \(1.10891\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{154} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 154,\ (\ :1/2),\ 0.868 - 0.494i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31659 + 0.348571i\)
\(L(\frac12)\) \(\approx\) \(1.31659 + 0.348571i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-1.50 - 2.95i)T \)
good3 \( 1 + (-1.89 + 1.38i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.865 - 2.66i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-2.18 + 6.72i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.850 + 2.61i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (6.81 - 4.94i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 3.83T + 23T^{2} \)
29 \( 1 + (3.75 + 2.72i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.0514 - 0.158i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.23 + 3.07i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.416 + 0.302i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 4.07T + 43T^{2} \)
47 \( 1 + (-7.15 + 5.19i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.399 - 1.23i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-5.80 - 4.21i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.422 - 1.29i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 3.87T + 67T^{2} \)
71 \( 1 + (0.872 + 2.68i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-5.11 - 3.71i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.227 - 0.700i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.99 - 15.3i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 2.18T + 89T^{2} \)
97 \( 1 + (-3.61 + 11.1i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.31192831994218480617005035682, −12.48196306847639346320301212318, −10.64059545746785755153533763121, −9.958508625599283857219105079504, −8.611105228899533293080881225233, −7.70846297852155539996011740409, −6.89847737514627141511643709728, −5.89762457389607825250173926974, −3.65897396694903842007938837651, −2.22846785621780543874870388604, 1.97429048964656077914805595663, 3.71663529730624689214710375134, 4.56517503748062788273818367528, 6.35060089117208595531779863505, 8.552320559656904489600062938873, 8.852131347445505335243441003788, 9.456367763939554003123767028149, 10.74914292833790719386683312216, 11.84599167245147658049854485807, 13.03764129157462753129089851906

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