Properties

Label 2-397800-1.1-c1-0-24
Degree $2$
Conductor $397800$
Sign $1$
Analytic cond. $3176.44$
Root an. cond. $56.3599$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·11-s + 13-s + 17-s + 8·19-s + 6·29-s − 8·31-s − 2·37-s + 10·41-s − 4·43-s − 7·49-s − 6·53-s − 6·61-s − 4·67-s + 8·71-s + 14·73-s + 4·79-s − 4·83-s + 14·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.20·11-s + 0.277·13-s + 0.242·17-s + 1.83·19-s + 1.11·29-s − 1.43·31-s − 0.328·37-s + 1.56·41-s − 0.609·43-s − 49-s − 0.824·53-s − 0.768·61-s − 0.488·67-s + 0.949·71-s + 1.63·73-s + 0.450·79-s − 0.439·83-s + 1.48·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(397800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(3176.44\)
Root analytic conductor: \(56.3599\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{397800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 397800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.132954951\)
\(L(\frac12)\) \(\approx\) \(2.132954951\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
13 \( 1 - T \)
17 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
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 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
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 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.41152650964142, −12.08467842100014, −11.47627375194830, −11.01779334920119, −10.74916546797157, −10.16528790825637, −9.728884671361294, −9.319458576147485, −8.919581197598048, −8.127434238201535, −7.938464290037674, −7.490280188872678, −7.032070000642881, −6.404357925121590, −5.935084418215063, −5.363041165971748, −5.047397917284498, −4.639445455019561, −3.782102717888791, −3.394368513358557, −2.895541524764665, −2.387413157533640, −1.664678679838317, −1.071116591123078, −0.4041618669864571, 0.4041618669864571, 1.071116591123078, 1.664678679838317, 2.387413157533640, 2.895541524764665, 3.394368513358557, 3.782102717888791, 4.639445455019561, 5.047397917284498, 5.363041165971748, 5.935084418215063, 6.404357925121590, 7.032070000642881, 7.490280188872678, 7.938464290037674, 8.127434238201535, 8.919581197598048, 9.319458576147485, 9.728884671361294, 10.16528790825637, 10.74916546797157, 11.01779334920119, 11.47627375194830, 12.08467842100014, 12.41152650964142

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