Properties

Label 2-154-1.1-c1-0-4
Degree $2$
Conductor $154$
Sign $-1$
Analytic cond. $1.22969$
Root an. cond. $1.10891$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 4·5-s − 7-s − 8-s − 3·9-s + 4·10-s − 11-s + 2·13-s + 14-s + 16-s − 4·17-s + 3·18-s − 6·19-s − 4·20-s + 22-s + 4·23-s + 11·25-s − 2·26-s − 28-s − 2·29-s − 2·31-s − 32-s + 4·34-s + 4·35-s − 3·36-s + 10·37-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.377·7-s − 0.353·8-s − 9-s + 1.26·10-s − 0.301·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.707·18-s − 1.37·19-s − 0.894·20-s + 0.213·22-s + 0.834·23-s + 11/5·25-s − 0.392·26-s − 0.188·28-s − 0.371·29-s − 0.359·31-s − 0.176·32-s + 0.685·34-s + 0.676·35-s − 1/2·36-s + 1.64·37-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(1.22969\)
Root analytic conductor: \(1.10891\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{154} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 154,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 + T \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−12.22063198281703280312046818474, −11.19149675055500729441636567526, −10.82425456930030561555637854181, −9.025591496790457491942383223256, −8.383894826086909083235343969228, −7.41620352139467519733544989343, −6.22872197038012242892904428945, −4.35107164128550817055960303512, −2.98342596760904293099474363551, 0, 2.98342596760904293099474363551, 4.35107164128550817055960303512, 6.22872197038012242892904428945, 7.41620352139467519733544989343, 8.383894826086909083235343969228, 9.025591496790457491942383223256, 10.82425456930030561555637854181, 11.19149675055500729441636567526, 12.22063198281703280312046818474

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