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Properties

Label 2-2496-312.269-c0-0-3
Degree $2$
Conductor $2496$
Sign $-0.862 - 0.505i$
Analytic cond. $1.24566$
Root an. cond. $1.11609$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)3-s + (−0.866 − 1.5i)7-s + (0.499 + 0.866i)9-s + (−0.5 − 0.866i)13-s + (−1.73 + i)19-s + 1.73i·21-s − 25-s − 0.999i·27-s + 1.73·31-s + 0.999i·39-s + (−0.866 + 0.5i)43-s + (−1 + 1.73i)49-s + 1.99·57-s + (−1.5 + 0.866i)61-s + (0.866 − 1.49i)63-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (−0.866 − 1.5i)7-s + (0.499 + 0.866i)9-s + (−0.5 − 0.866i)13-s + (−1.73 + i)19-s + 1.73i·21-s − 25-s − 0.999i·27-s + 1.73·31-s + 0.999i·39-s + (−0.866 + 0.5i)43-s + (−1 + 1.73i)49-s + 1.99·57-s + (−1.5 + 0.866i)61-s + (0.866 − 1.49i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $-0.862 - 0.505i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :0),\ -0.862 - 0.505i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1593021923\)
\(L(\frac12)\) \(\approx\) \(0.1593021923\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + T^{2} \)
7 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - 1.73T + T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + 1.73T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.377353032870787846956742156497, −7.77483839416767934073321876544, −7.03856806834126905778992411311, −6.35807048283327852989343474432, −5.78907820465172595423367934262, −4.57667652130147560354973427061, −4.00996796064118431811712398268, −2.81067389397980219954169558876, −1.43956329432320957793261525217, −0.11564035946231940781869899568, 2.03752413813980219438026467152, 2.95565617498756999996891506069, 4.16737235097695936825130720648, 4.84742946412790409797788299212, 5.75581099662417679375371759055, 6.43183305306719711413274713605, 6.83222986379958975875993946997, 8.236358770722923120519154284790, 8.999229490629660682373248221017, 9.517857692147290943878269628919

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