Properties

Label 2-2496-1.1-c1-0-6
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 − 2·7-s + 9-s + 4·11-s − 13-s − 6·17-s + 6·19-s + 2·21-s − 5·25-s − 27-s + 2·29-s + 6·31-s − 4·33-s − 10·37-s + 39-s + 8·41-s + 12·43-s − 12·47-s − 3·49-s + 6·51-s + 6·53-s − 6·57-s − 2·61-s − 2·63-s + 2·67-s + 8·71-s + 14·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 1.45·17-s + 1.37·19-s + 0.436·21-s − 25-s − 0.192·27-s + 0.371·29-s + 1.07·31-s − 0.696·33-s − 1.64·37-s + 0.160·39-s + 1.24·41-s + 1.82·43-s − 1.75·47-s − 3/7·49-s + 0.840·51-s + 0.824·53-s − 0.794·57-s − 0.256·61-s − 0.251·63-s + 0.244·67-s + 0.949·71-s + 1.63·73-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: $\chi_{2496} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.268600395\)
\(L(\frac12)\) \(\approx\) \(1.268600395\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−9.220283570203917667028022793155, −8.137716429689322885874117903176, −7.16333848096912753102386607562, −6.57091938769509480942354725697, −5.98247083912518480972422452649, −4.97254580700680097900229302596, −4.13958008081477749813044275306, −3.29289095796707662296197413885, −2.06399639050261723146787386632, −0.73510269161810698907992625597, 0.73510269161810698907992625597, 2.06399639050261723146787386632, 3.29289095796707662296197413885, 4.13958008081477749813044275306, 4.97254580700680097900229302596, 5.98247083912518480972422452649, 6.57091938769509480942354725697, 7.16333848096912753102386607562, 8.137716429689322885874117903176, 9.220283570203917667028022793155

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