Properties

Label 2-154-11.9-c1-0-0
Degree $2$
Conductor $154$
Sign $-0.437 - 0.899i$
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 + (−0.5 + 0.363i)3-s + (−0.809 − 0.587i)4-s + (0.618 + 1.90i)5-s + (−0.190 − 0.587i)6-s + (0.809 + 0.587i)7-s + (0.809 − 0.587i)8-s + (−0.809 + 2.48i)9-s − 1.99·10-s + (−3.04 + 1.31i)11-s + 0.618·12-s + (−0.381 + 1.17i)13-s + (−0.809 + 0.587i)14-s + (−1 − 0.726i)15-s + (0.309 + 0.951i)16-s + (0.263 + 0.812i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.288 + 0.209i)3-s + (−0.404 − 0.293i)4-s + (0.276 + 0.850i)5-s + (−0.0779 − 0.239i)6-s + (0.305 + 0.222i)7-s + (0.286 − 0.207i)8-s + (−0.269 + 0.829i)9-s − 0.632·10-s + (−0.918 + 0.396i)11-s + 0.178·12-s + (−0.105 + 0.326i)13-s + (−0.216 + 0.157i)14-s + (−0.258 − 0.187i)15-s + (0.0772 + 0.237i)16-s + (0.0640 + 0.197i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $-0.437 - 0.899i$
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: \((2,\ 154,\ (\ :1/2),\ -0.437 - 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.466298 + 0.745770i\)
\(L(\frac12)\) \(\approx\) \(0.466298 + 0.745770i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.04 - 1.31i)T \)
good3 \( 1 + (0.5 - 0.363i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.618 - 1.90i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (0.381 - 1.17i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.263 - 0.812i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-5.54 + 4.02i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 0.472T + 23T^{2} \)
29 \( 1 + (-6.47 - 4.70i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.38 + 4.25i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.61 + 3.35i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.73 + 3.44i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 1.85T + 43T^{2} \)
47 \( 1 + (-9.47 + 6.88i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (4 - 12.3i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.78 + 6.37i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.763 + 2.35i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 + (-0.472 - 1.45i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.73 - 2.71i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1 + 3.07i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.95 - 15.2i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 0.326T + 89T^{2} \)
97 \( 1 + (3.04 - 9.37i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.70207990341289899046647921606, −12.26973755948007580242539111638, −10.94619296098511763329752998824, −10.37879560633075023694019842448, −9.169739219638960014396150217635, −7.87811550807892099801914168491, −7.02055330562513519206322120720, −5.69030176823974327248486304361, −4.75582238432853706319910556332, −2.60374388067666243876877972287, 1.04725796057058880884171440520, 3.13571364566150972936328767804, 4.82990760273024257856527022806, 5.90260068799733650700026680225, 7.62677702051659098767023098909, 8.626032926727745947114646897389, 9.666847599135427447376037989977, 10.63220161376494828459532161445, 11.83313498165413430350901469402, 12.41652942329558184710390647938

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