Properties

Label 2-138-1.1-c7-0-14
Degree $2$
Conductor $138$
Sign $1$
Analytic cond. $43.1091$
Root an. cond. $6.56575$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s + 27·3-s + 64·4-s − 10.0·5-s + 216·6-s + 971.·7-s + 512·8-s + 729·9-s − 80.5·10-s − 2.06e3·11-s + 1.72e3·12-s + 9.53e3·13-s + 7.76e3·14-s − 271.·15-s + 4.09e3·16-s − 2.08e4·17-s + 5.83e3·18-s + 4.10e4·19-s − 644.·20-s + 2.62e4·21-s − 1.65e4·22-s − 1.21e4·23-s + 1.38e4·24-s − 7.80e4·25-s + 7.62e4·26-s + 1.96e4·27-s + 6.21e4·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.0360·5-s + 0.408·6-s + 1.07·7-s + 0.353·8-s + 0.333·9-s − 0.0254·10-s − 0.468·11-s + 0.288·12-s + 1.20·13-s + 0.756·14-s − 0.0207·15-s + 0.250·16-s − 1.02·17-s + 0.235·18-s + 1.37·19-s − 0.0180·20-s + 0.617·21-s − 0.331·22-s − 0.208·23-s + 0.204·24-s − 0.998·25-s + 0.851·26-s + 0.192·27-s + 0.535·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $1$
Analytic conductor: \(43.1091\)
Root analytic conductor: \(6.56575\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(4.782560852\)
\(L(\frac12)\) \(\approx\) \(4.782560852\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 8T \)
3 \( 1 - 27T \)
23 \( 1 + 1.21e4T \)
good5 \( 1 + 10.0T + 7.81e4T^{2} \)
7 \( 1 - 971.T + 8.23e5T^{2} \)
11 \( 1 + 2.06e3T + 1.94e7T^{2} \)
13 \( 1 - 9.53e3T + 6.27e7T^{2} \)
17 \( 1 + 2.08e4T + 4.10e8T^{2} \)
19 \( 1 - 4.10e4T + 8.93e8T^{2} \)
29 \( 1 - 2.46e4T + 1.72e10T^{2} \)
31 \( 1 - 2.44e5T + 2.75e10T^{2} \)
37 \( 1 - 5.20e5T + 9.49e10T^{2} \)
41 \( 1 - 1.23e5T + 1.94e11T^{2} \)
43 \( 1 - 7.42e3T + 2.71e11T^{2} \)
47 \( 1 + 1.38e6T + 5.06e11T^{2} \)
53 \( 1 - 1.01e6T + 1.17e12T^{2} \)
59 \( 1 - 2.93e6T + 2.48e12T^{2} \)
61 \( 1 + 2.36e6T + 3.14e12T^{2} \)
67 \( 1 - 3.19e6T + 6.06e12T^{2} \)
71 \( 1 + 3.49e6T + 9.09e12T^{2} \)
73 \( 1 - 2.48e6T + 1.10e13T^{2} \)
79 \( 1 + 4.80e6T + 1.92e13T^{2} \)
83 \( 1 - 7.77e6T + 2.71e13T^{2} \)
89 \( 1 - 5.39e5T + 4.42e13T^{2} \)
97 \( 1 + 1.04e7T + 8.07e13T^{2} \)
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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

−11.73807394350758679866697110661, −11.13496336057619076193984703502, −9.839549731751206488044019576977, −8.433469927212164061211520789675, −7.70633699156495431572487492622, −6.29134115546031098007449671901, −4.99547553315877963488143416291, −3.92244545264270325534484283473, −2.55535905929611009953205279110, −1.25749473667986984974774172860, 1.25749473667986984974774172860, 2.55535905929611009953205279110, 3.92244545264270325534484283473, 4.99547553315877963488143416291, 6.29134115546031098007449671901, 7.70633699156495431572487492622, 8.433469927212164061211520789675, 9.839549731751206488044019576977, 11.13496336057619076193984703502, 11.73807394350758679866697110661

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