Properties

Label 2-2496-1.1-c1-0-29
Degree $2$
Conductor $2496$
Sign $-1$
Analytic cond. $19.9306$
Root an. cond. $4.46437$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s + 9-s − 13-s + 2·15-s + 2·17-s + 4·19-s − 25-s − 27-s − 6·29-s + 2·37-s + 39-s + 6·41-s + 12·43-s − 2·45-s − 4·47-s − 7·49-s − 2·51-s − 6·53-s − 4·57-s + 8·59-s + 2·61-s + 2·65-s − 4·67-s − 12·71-s − 14·73-s + 75-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.277·13-s + 0.516·15-s + 0.485·17-s + 0.917·19-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.328·37-s + 0.160·39-s + 0.937·41-s + 1.82·43-s − 0.298·45-s − 0.583·47-s − 49-s − 0.280·51-s − 0.824·53-s − 0.529·57-s + 1.04·59-s + 0.256·61-s + 0.248·65-s − 0.488·67-s − 1.42·71-s − 1.63·73-s + 0.115·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $-1$
Analytic conductor: \(19.9306\)
Root analytic conductor: \(4.46437\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2496,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
13 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−8.429645114873483352252590672860, −7.51727934813324433491134862469, −7.30784619649759594552263736405, −6.08537646602212143968327685330, −5.46962489281230822667740001129, −4.49909620784530543421734041489, −3.79070284272267126319936744658, −2.78518661391770994956776280001, −1.32676341715050635172211285974, 0, 1.32676341715050635172211285974, 2.78518661391770994956776280001, 3.79070284272267126319936744658, 4.49909620784530543421734041489, 5.46962489281230822667740001129, 6.08537646602212143968327685330, 7.30784619649759594552263736405, 7.51727934813324433491134862469, 8.429645114873483352252590672860

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