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Properties

Label 2-2156-11.10-c0-0-5
Degree $2$
Conductor $2156$
Sign $1$
Analytic cond. $1.07598$
Root an. cond. $1.03729$
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  + 3-s + 5-s + 11-s + 15-s − 23-s − 27-s + 31-s + 33-s − 37-s − 2·47-s + 2·53-s + 55-s + 59-s − 67-s − 69-s − 71-s − 81-s + 89-s + 93-s + 97-s − 2·103-s − 111-s − 113-s − 115-s + ⋯
L(s)  = 1  + 3-s + 5-s + 11-s + 15-s − 23-s − 27-s + 31-s + 33-s − 37-s − 2·47-s + 2·53-s + 55-s + 59-s − 67-s − 69-s − 71-s − 81-s + 89-s + 93-s + 97-s − 2·103-s − 111-s − 113-s − 115-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2156\)    =    \(2^{2} \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(1.07598\)
Root analytic conductor: \(1.03729\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2156} (197, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2156,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.916557754\)
\(L(\frac12)\) \(\approx\) \(1.916557754\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.210701067370210101002090936440, −8.596855369050174302178726013670, −7.926063135390894925876445683631, −6.88673750241902464825793637545, −6.16392564512325682840180943059, −5.40221853511302549423410898486, −4.22598581134903796482425926069, −3.38634873319861200844378269917, −2.39216894609072984070313684475, −1.58972073244680755357209865646, 1.58972073244680755357209865646, 2.39216894609072984070313684475, 3.38634873319861200844378269917, 4.22598581134903796482425926069, 5.40221853511302549423410898486, 6.16392564512325682840180943059, 6.88673750241902464825793637545, 7.926063135390894925876445683631, 8.596855369050174302178726013670, 9.210701067370210101002090936440

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