Properties

Label 2-13923-1.1-c1-0-10
Degree $2$
Conductor $13923$
Sign $1$
Analytic cond. $111.175$
Root an. cond. $10.5439$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 2·5-s + 7-s + 3·8-s + 2·10-s − 13-s − 14-s − 16-s − 17-s − 8·19-s + 2·20-s − 6·23-s − 25-s + 26-s − 28-s − 6·29-s − 8·31-s − 5·32-s + 34-s − 2·35-s − 4·37-s + 8·38-s − 6·40-s − 6·41-s + 6·46-s − 8·47-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.894·5-s + 0.377·7-s + 1.06·8-s + 0.632·10-s − 0.277·13-s − 0.267·14-s − 1/4·16-s − 0.242·17-s − 1.83·19-s + 0.447·20-s − 1.25·23-s − 1/5·25-s + 0.196·26-s − 0.188·28-s − 1.11·29-s − 1.43·31-s − 0.883·32-s + 0.171·34-s − 0.338·35-s − 0.657·37-s + 1.29·38-s − 0.948·40-s − 0.937·41-s + 0.884·46-s − 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13923 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13923 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13923\)    =    \(3^{2} \cdot 7 \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(111.175\)
Root analytic conductor: \(10.5439\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{13923} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 13923,\ (\ :1/2),\ 1)\)

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)$
bad3 \( 1 \)
7 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−16.92215597109170, −16.27961844479419, −15.65119677164086, −15.00689889396718, −14.62760631097673, −13.97034848105212, −13.29741684189543, −12.76402809221014, −12.21816278531404, −11.46959536222055, −11.00370783721477, −10.39290578288956, −9.879213095011560, −9.076064839529302, −8.639017166989239, −8.068080363927691, −7.618796714050987, −6.999094282706610, −6.149177903872702, −5.321048762039072, −4.601424168435247, −4.006638857618852, −3.551221440089895, −2.142698355661508, −1.619163511506154, 0, 0, 1.619163511506154, 2.142698355661508, 3.551221440089895, 4.006638857618852, 4.601424168435247, 5.321048762039072, 6.149177903872702, 6.999094282706610, 7.618796714050987, 8.068080363927691, 8.639017166989239, 9.076064839529302, 9.879213095011560, 10.39290578288956, 11.00370783721477, 11.46959536222055, 12.21816278531404, 12.76402809221014, 13.29741684189543, 13.97034848105212, 14.62760631097673, 15.00689889396718, 15.65119677164086, 16.27961844479419, 16.92215597109170

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