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Properties

Label 2-34e2-4.3-c0-0-2
Degree $2$
Conductor $1156$
Sign $1$
Analytic cond. $0.576919$
Root an. cond. $0.759551$
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 − 2·13-s + 16-s + 18-s − 25-s − 2·26-s + 32-s + 36-s + 49-s − 50-s − 2·52-s − 2·53-s + 64-s + 72-s + 81-s − 2·89-s + 98-s − 100-s − 2·101-s − 2·104-s − 2·106-s − 2·117-s + ⋯
L(s)  = 1  + 2-s + 4-s + 8-s + 9-s − 2·13-s + 16-s + 18-s − 25-s − 2·26-s + 32-s + 36-s + 49-s − 50-s − 2·52-s − 2·53-s + 64-s + 72-s + 81-s − 2·89-s + 98-s − 100-s − 2·101-s − 2·104-s − 2·106-s − 2·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1156} (579, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.946331439\)
\(L(\frac12)\) \(\approx\) \(1.946331439\)
\(L(1)\) 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 \( 1 \)
good3 \( ( 1 - T )( 1 + T ) \)
5 \( 1 + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 + T )^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( 1 + T^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( 1 + T^{2} \)
41 \( 1 + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 + T )^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 + T )^{2} \)
97 \( 1 + T^{2} \)
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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

−10.00459735063275287076119707895, −9.509310261511122666221200381479, −7.979954141971087890146568066000, −7.34390078513092637331110970460, −6.68936916288307072353651932438, −5.58480636570868322555035297652, −4.75228434759572436676809041751, −4.05990622505679150187364631673, −2.81987650161538688926266694129, −1.81128345542631433367044257082, 1.81128345542631433367044257082, 2.81987650161538688926266694129, 4.05990622505679150187364631673, 4.75228434759572436676809041751, 5.58480636570868322555035297652, 6.68936916288307072353651932438, 7.34390078513092637331110970460, 7.979954141971087890146568066000, 9.509310261511122666221200381479, 10.00459735063275287076119707895

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