↑

Properties

Label 2-7938-1.1-c1-0-66
Degree $2$
Conductor $7938$
Sign $-1$
Analytic cond. $63.3852$
Root an. cond. $7.96148$
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-s + 4-s − 4·5-s − 8-s + 4·10-s − 2·11-s + 6·13-s + 16-s − 2·17-s − 4·19-s − 4·20-s + 2·22-s + 23-s + 11·25-s − 6·26-s − 4·29-s − 9·31-s − 32-s + 2·34-s + 8·37-s + 4·38-s + 4·40-s − 3·41-s + 2·43-s − 2·44-s − 46-s + 9·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.353·8-s + 1.26·10-s − 0.603·11-s + 1.66·13-s + 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.894·20-s + 0.426·22-s + 0.208·23-s + 11/5·25-s − 1.17·26-s − 0.742·29-s − 1.61·31-s − 0.176·32-s + 0.342·34-s + 1.31·37-s + 0.648·38-s + 0.632·40-s − 0.468·41-s + 0.304·43-s − 0.301·44-s − 0.147·46-s + 1.31·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7938\)    =    \(2 \cdot 3^{4} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(63.3852\)
Root analytic conductor: \(7.96148\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7938,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
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

−7.53609730254760862737794881919, −7.14488072434730187809108518690, −6.24742797454429626132925213363, −5.53483124209901011456809089552, −4.38636601925757611843209560043, −3.88697259602970350610614709456, −3.20814137194403173780383720362, −2.15703073734121508276907104381, −0.940698283034662263787948484209, 0, 0.940698283034662263787948484209, 2.15703073734121508276907104381, 3.20814137194403173780383720362, 3.88697259602970350610614709456, 4.38636601925757611843209560043, 5.53483124209901011456809089552, 6.24742797454429626132925213363, 7.14488072434730187809108518690, 7.53609730254760862737794881919

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