Properties

Label 2-10304-1.1-c1-0-10
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 − 2·17-s − 2·21-s + 23-s − 25-s + 4·27-s + 2·29-s + 4·33-s − 2·35-s + 4·37-s + 8·39-s + 6·41-s − 2·43-s − 2·45-s + 4·47-s + 49-s + 4·51-s + 4·55-s − 2·59-s + 10·61-s + 63-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 − 0.485·17-s − 0.436·21-s + 0.208·23-s − 1/5·25-s + 0.769·27-s + 0.371·29-s + 0.696·33-s − 0.338·35-s + 0.657·37-s + 1.28·39-s + 0.937·41-s − 0.304·43-s − 0.298·45-s + 0.583·47-s + 1/7·49-s + 0.560·51-s + 0.539·55-s − 0.260·59-s + 1.28·61-s + 0.125·63-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 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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

−16.91208332511826, −16.36244137166378, −15.73058644092489, −15.40485093080377, −14.57061407906520, −14.23265205943321, −13.16773763213170, −12.78469027864833, −12.01115766788416, −11.67278362934388, −11.25602391923793, −10.50403052934678, −10.14544997134552, −9.186398647822894, −8.536017817456366, −7.675937734443965, −7.440212654826769, −6.559171157434498, −5.903646284859285, −5.149935606091350, −4.686139442532922, −4.039947963096991, −2.963589363880053, −2.179383177492332, −0.8213983700428921, 0, 0.8213983700428921, 2.179383177492332, 2.963589363880053, 4.039947963096991, 4.686139442532922, 5.149935606091350, 5.903646284859285, 6.559171157434498, 7.440212654826769, 7.675937734443965, 8.536017817456366, 9.186398647822894, 10.14544997134552, 10.50403052934678, 11.25602391923793, 11.67278362934388, 12.01115766788416, 12.78469027864833, 13.16773763213170, 14.23265205943321, 14.57061407906520, 15.40485093080377, 15.73058644092489, 16.36244137166378, 16.91208332511826

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