Properties

Label 2-10304-1.1-c1-0-36
Degree $2$
Conductor $10304$
Sign $-1$
Analytic cond. $82.2778$
Root an. cond. $9.07071$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·5-s + 7-s + 9-s + 2·11-s − 4·13-s + 4·15-s − 6·17-s − 4·19-s + 2·21-s − 23-s − 25-s − 4·27-s − 6·29-s − 4·31-s + 4·33-s + 2·35-s + 8·37-s − 8·39-s + 6·41-s − 6·43-s + 2·45-s − 8·47-s + 49-s − 12·51-s + 4·53-s + 4·55-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s + 1.03·15-s − 1.45·17-s − 0.917·19-s + 0.436·21-s − 0.208·23-s − 1/5·25-s − 0.769·27-s − 1.11·29-s − 0.718·31-s + 0.696·33-s + 0.338·35-s + 1.31·37-s − 1.28·39-s + 0.937·41-s − 0.914·43-s + 0.298·45-s − 1.16·47-s + 1/7·49-s − 1.68·51-s + 0.549·53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10304\)    =    \(2^{6} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(82.2778\)
Root analytic conductor: \(9.07071\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10304} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 10304,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
23 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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

−16.92489068434041, −16.51337609938092, −15.31223516512347, −15.11723611694062, −14.58048004526530, −14.02807684846854, −13.60986041829871, −12.92662322615946, −12.56628760631929, −11.45896578405788, −11.18570294227953, −10.28494408073950, −9.571884607620886, −9.239457052549565, −8.749120720062985, −7.944998771598409, −7.478549307776074, −6.581756466965419, −6.071100541737439, −5.183075198944190, −4.411093756540659, −3.795546442393571, −2.761173194431555, −2.136941713222546, −1.722510272521193, 0, 1.722510272521193, 2.136941713222546, 2.761173194431555, 3.795546442393571, 4.411093756540659, 5.183075198944190, 6.071100541737439, 6.581756466965419, 7.478549307776074, 7.944998771598409, 8.749120720062985, 9.239457052549565, 9.571884607620886, 10.28494408073950, 11.18570294227953, 11.45896578405788, 12.56628760631929, 12.92662322615946, 13.60986041829871, 14.02807684846854, 14.58048004526530, 15.11723611694062, 15.31223516512347, 16.51337609938092, 16.92489068434041

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