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Properties

Label 2-2275-91.90-c0-0-4
Degree $2$
Conductor $2275$
Sign $1$
Analytic cond. $1.13537$
Root an. cond. $1.06553$
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  + 4-s − 7-s + 9-s + 13-s + 16-s − 28-s − 2·29-s + 36-s + 2·47-s + 49-s + 52-s − 63-s + 64-s + 2·73-s − 2·79-s + 81-s − 2·83-s − 91-s − 2·97-s − 112-s − 2·116-s + 117-s + ⋯
L(s)  = 1  + 4-s − 7-s + 9-s + 13-s + 16-s − 28-s − 2·29-s + 36-s + 2·47-s + 49-s + 52-s − 63-s + 64-s + 2·73-s − 2·79-s + 81-s − 2·83-s − 91-s − 2·97-s − 112-s − 2·116-s + 117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2275\)    =    \(5^{2} \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(1.13537\)
Root analytic conductor: \(1.06553\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2275} (2001, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2275,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.520655541\)
\(L(\frac12)\) \(\approx\) \(1.520655541\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + T \)
13 \( 1 - T \)
good2 \( ( 1 - T )( 1 + T ) \)
3 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( ( 1 + T )^{2} \)
31 \( 1 + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( ( 1 - T )^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )^{2} \)
79 \( ( 1 + T )^{2} \)
83 \( ( 1 + T )^{2} \)
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

−9.350893980705210654345198599649, −8.404490565352293145703996780486, −7.35145893123702314437887175603, −7.02114443020539493347739410631, −6.11073436853892686369561492075, −5.56887069264097216609209273935, −4.09107571139056533747939320439, −3.49555542768486508026115938405, −2.40924272592946609760302468047, −1.32852491915568969095415422707, 1.32852491915568969095415422707, 2.40924272592946609760302468047, 3.49555542768486508026115938405, 4.09107571139056533747939320439, 5.56887069264097216609209273935, 6.11073436853892686369561492075, 7.02114443020539493347739410631, 7.35145893123702314437887175603, 8.404490565352293145703996780486, 9.350893980705210654345198599649

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