Properties

Label 2-10304-1.1-c1-0-2
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 $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4·5-s − 7-s + 9-s − 4·11-s − 4·13-s − 8·15-s − 6·17-s − 2·19-s + 2·21-s − 23-s + 11·25-s + 4·27-s + 6·29-s − 8·31-s + 8·33-s − 4·35-s − 10·37-s + 8·39-s + 6·41-s − 12·43-s + 4·45-s − 8·47-s + 49-s + 12·51-s + 2·53-s − 16·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s − 2.06·15-s − 1.45·17-s − 0.458·19-s + 0.436·21-s − 0.208·23-s + 11/5·25-s + 0.769·27-s + 1.11·29-s − 1.43·31-s + 1.39·33-s − 0.676·35-s − 1.64·37-s + 1.28·39-s + 0.937·41-s − 1.82·43-s + 0.596·45-s − 1.16·47-s + 1/7·49-s + 1.68·51-s + 0.274·53-s − 2.15·55-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: \(0\)
Selberg data: \((2,\ 10304,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7790913405\)
\(L(\frac12)\) \(\approx\) \(0.7790913405\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−16.77189518175945, −16.14888090781012, −15.67042685968009, −14.76223221184689, −14.31578011375882, −13.53904277667516, −13.04136685799007, −12.74966633150979, −12.00134088378267, −11.24699596517115, −10.64149769776967, −10.13662064621562, −9.884915949099752, −8.911941807173745, −8.511765923993644, −7.284978689495105, −6.716458891782044, −6.252302893636411, −5.547821197319770, −5.071700649220687, −4.651630461559484, −3.198012833660136, −2.324480867427205, −1.893354316761040, −0.4111692734705891, 0.4111692734705891, 1.893354316761040, 2.324480867427205, 3.198012833660136, 4.651630461559484, 5.071700649220687, 5.547821197319770, 6.252302893636411, 6.716458891782044, 7.284978689495105, 8.511765923993644, 8.911941807173745, 9.884915949099752, 10.13662064621562, 10.64149769776967, 11.24699596517115, 12.00134088378267, 12.74966633150979, 13.04136685799007, 13.53904277667516, 14.31578011375882, 14.76223221184689, 15.67042685968009, 16.14888090781012, 16.77189518175945

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