Properties

Label 2-85176-1.1-c1-0-9
Degree $2$
Conductor $85176$
Sign $1$
Analytic cond. $680.133$
Root an. cond. $26.0793$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 7-s − 4·11-s + 6·17-s − 8·19-s − 25-s − 6·29-s − 8·31-s + 2·35-s + 2·37-s + 2·41-s − 4·43-s − 8·47-s + 49-s − 6·53-s − 8·55-s − 6·61-s + 4·67-s − 8·71-s − 10·73-s − 4·77-s + 16·79-s + 8·83-s + 12·85-s − 6·89-s − 16·95-s + 6·97-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s − 1.20·11-s + 1.45·17-s − 1.83·19-s − 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.338·35-s + 0.328·37-s + 0.312·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 1.07·55-s − 0.768·61-s + 0.488·67-s − 0.949·71-s − 1.17·73-s − 0.455·77-s + 1.80·79-s + 0.878·83-s + 1.30·85-s − 0.635·89-s − 1.64·95-s + 0.609·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85176\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(680.133\)
Root analytic conductor: \(26.0793\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{85176} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 85176,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.470752405\)
\(L(\frac12)\) \(\approx\) \(1.470752405\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
13 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.94427954008100, −13.33685481177663, −12.85099079105821, −12.75280596596686, −11.99909832713037, −11.32990742650406, −10.89147368309974, −10.35235596811974, −10.10042840671862, −9.371557844804988, −9.052032100070287, −8.254909063668460, −7.844418889519811, −7.499321549630342, −6.668365327764732, −6.126622420787198, −5.600904244019957, −5.251960179855980, −4.631153178042096, −3.898614617041553, −3.267452436863713, −2.577407135010646, −1.873216436962159, −1.596405254158065, −0.3624220808841593, 0.3624220808841593, 1.596405254158065, 1.873216436962159, 2.577407135010646, 3.267452436863713, 3.898614617041553, 4.631153178042096, 5.251960179855980, 5.600904244019957, 6.126622420787198, 6.668365327764732, 7.499321549630342, 7.844418889519811, 8.254909063668460, 9.052032100070287, 9.371557844804988, 10.10042840671862, 10.35235596811974, 10.89147368309974, 11.32990742650406, 11.99909832713037, 12.75280596596686, 12.85099079105821, 13.33685481177663, 13.94427954008100

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