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Properties

Label 2-154-11.9-c1-0-4
Degree $2$
Conductor $154$
Sign $0.263 + 0.964i$
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 + (1.80 − 1.31i)3-s + (−0.809 − 0.587i)4-s + (0.190 + 0.587i)5-s + (−0.690 − 2.12i)6-s + (0.809 + 0.587i)7-s + (−0.809 + 0.587i)8-s + (0.618 − 1.90i)9-s + 0.618·10-s + (−3.23 + 0.726i)11-s − 2.23·12-s + (−0.0729 + 0.224i)13-s + (0.809 − 0.587i)14-s + (1.11 + 0.812i)15-s + (0.309 + 0.951i)16-s + (−0.0729 − 0.224i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (1.04 − 0.758i)3-s + (−0.404 − 0.293i)4-s + (0.0854 + 0.262i)5-s + (−0.282 − 0.868i)6-s + (0.305 + 0.222i)7-s + (−0.286 + 0.207i)8-s + (0.206 − 0.634i)9-s + 0.195·10-s + (−0.975 + 0.219i)11-s − 0.645·12-s + (−0.0202 + 0.0622i)13-s + (0.216 − 0.157i)14-s + (0.288 + 0.209i)15-s + (0.0772 + 0.237i)16-s + (−0.0176 − 0.0544i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $0.263 + 0.964i$
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: \((2,\ 154,\ (\ :1/2),\ 0.263 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24742 - 0.952423i\)
\(L(\frac12)\) \(\approx\) \(1.24742 - 0.952423i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.23 - 0.726i)T \)
good3 \( 1 + (-1.80 + 1.31i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.190 - 0.587i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (0.0729 - 0.224i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.0729 + 0.224i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.04 - 2.93i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 0.236T + 23T^{2} \)
29 \( 1 + (-1.19 - 0.865i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.69 + 8.28i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.35 - 4.61i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (7.23 - 5.25i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.85T + 43T^{2} \)
47 \( 1 + (5.35 - 3.88i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.54 + 7.83i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (6.97 + 5.06i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.64 - 5.06i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 9.18T + 67T^{2} \)
71 \( 1 + (4.16 + 12.8i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (7.78 + 5.65i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.63 + 14.2i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.66 + 8.19i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 2.90T + 89T^{2} \)
97 \( 1 + (4.14 - 12.7i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.99430950867237247979065458028, −11.93932125161084455591283156165, −10.76019721135477296475493493365, −9.768813697802095935879720351548, −8.458650517613309328045257515364, −7.82498616215645225941814685665, −6.35026282364664994657034893995, −4.72600884977557821452743814397, −3.01991142758810425494050635641, −2.00078743378266391681298820411, 2.86468475339095836054132156449, 4.25374700624606249348026775320, 5.30204351657925732967452883599, 6.92166967870184440028534078895, 8.240371914129620570905968267444, 8.767904031450825715850771662249, 9.933170430434913165872331743899, 10.93014585879814685281128072419, 12.53944287396798677881412591244, 13.47320024201156079304134767806

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