Properties

Label 2-2496-1.1-c1-0-0
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s − 4·7-s + 9-s − 4·11-s − 13-s + 2·15-s + 2·17-s − 8·19-s + 4·21-s − 25-s − 27-s − 6·29-s + 4·31-s + 4·33-s + 8·35-s + 2·37-s + 39-s − 10·41-s + 4·43-s − 2·45-s − 8·47-s + 9·49-s − 2·51-s + 10·53-s + 8·55-s + 8·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 0.516·15-s + 0.485·17-s − 1.83·19-s + 0.872·21-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.696·33-s + 1.35·35-s + 0.328·37-s + 0.160·39-s − 1.56·41-s + 0.609·43-s − 0.298·45-s − 1.16·47-s + 9/7·49-s − 0.280·51-s + 1.37·53-s + 1.07·55-s + 1.05·57-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: \(0\)
Selberg data: \((2,\ 2496,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2813098118\)
\(L(\frac12)\) \(\approx\) \(0.2813098118\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−8.885249003422558585437324551125, −8.056279462247341518969959407951, −7.36246990708953084574462420818, −6.56898921303829458916908478753, −5.92223885047057182378725074778, −4.99526176455710672796958116238, −4.04803186534857831005498278724, −3.30123270809335987982157307983, −2.25832575358208630780772683335, −0.32122777317457664777326917747, 0.32122777317457664777326917747, 2.25832575358208630780772683335, 3.30123270809335987982157307983, 4.04803186534857831005498278724, 4.99526176455710672796958116238, 5.92223885047057182378725074778, 6.56898921303829458916908478753, 7.36246990708953084574462420818, 8.056279462247341518969959407951, 8.885249003422558585437324551125

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