Properties

Label 2-136-136.67-c0-0-0
Degree $2$
Conductor $136$
Sign $1$
Analytic cond. $0.0678728$
Root an. cond. $0.260524$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s + 9-s + 16-s − 17-s − 18-s − 2·19-s − 25-s − 32-s + 34-s + 36-s + 2·38-s + 2·43-s − 49-s + 50-s + 2·59-s + 64-s − 2·67-s − 68-s − 72-s − 2·76-s + 81-s + 2·83-s − 2·86-s − 2·89-s + 98-s + ⋯
L(s)  = 1  − 2-s + 4-s − 8-s + 9-s + 16-s − 17-s − 18-s − 2·19-s − 25-s − 32-s + 34-s + 36-s + 2·38-s + 2·43-s − 49-s + 50-s + 2·59-s + 64-s − 2·67-s − 68-s − 72-s − 2·76-s + 81-s + 2·83-s − 2·86-s − 2·89-s + 98-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $1$
Analytic conductor: \(0.0678728\)
Root analytic conductor: \(0.260524\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{136} (67, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4342953204\)
\(L(\frac12)\) \(\approx\) \(0.4342953204\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−13.24083381227856928189287299030, −12.37060503711264230626462419909, −11.12990696280955116118199910364, −10.34652595840034658900516186202, −9.309527469753088104438031600551, −8.296626719201967746883695806752, −7.14365748685889483187532804768, −6.16344603975208278087909120617, −4.20651178016227437261911449698, −2.11561848960962061958698046486, 2.11561848960962061958698046486, 4.20651178016227437261911449698, 6.16344603975208278087909120617, 7.14365748685889483187532804768, 8.296626719201967746883695806752, 9.309527469753088104438031600551, 10.34652595840034658900516186202, 11.12990696280955116118199910364, 12.37060503711264230626462419909, 13.24083381227856928189287299030

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