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Properties

Label 2-2496-312.77-c0-0-10
Degree $2$
Conductor $2496$
Sign $0.258 + 0.965i$
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 − 1.73i·5-s − i·7-s + (0.499 + 0.866i)9-s − 13-s + (0.866 − 1.49i)15-s − 1.73i·17-s + (0.5 − 0.866i)21-s − 1.99·25-s + 0.999i·27-s + 2i·31-s − 1.73·35-s − 37-s + (−0.866 − 0.5i)39-s − i·43-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s − 1.73i·5-s − i·7-s + (0.499 + 0.866i)9-s − 13-s + (0.866 − 1.49i)15-s − 1.73i·17-s + (0.5 − 0.866i)21-s − 1.99·25-s + 0.999i·27-s + 2i·31-s − 1.73·35-s − 37-s + (−0.866 − 0.5i)39-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $0.258 + 0.965i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :0),\ 0.258 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.495194240\)
\(L(\frac12)\) \(\approx\) \(1.495194240\)
\(L(1)\) 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 \( 1 + T \)
good5 \( 1 + 1.73iT - T^{2} \)
7 \( 1 + iT - T^{2} \)
11 \( 1 - T^{2} \)
17 \( 1 + 1.73iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 2iT - T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 - 1.73T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - 1.73T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.925658111687465769317672096444, −8.418522654552514420257016930099, −7.40746379880658791956358306233, −7.09067431879173514676389608585, −5.32507478036261321330825551037, −4.92999958330322468280042980552, −4.26984671804894284111664767851, −3.34025781057296799867394903593, −2.14504466341566549480543018648, −0.892859957732629440490244545981, 2.05069323618113212242325656682, 2.46656753181396512134880143518, 3.36800298639933367475709975533, 4.17150696606773043674063541544, 5.72699816207724817138936451661, 6.27555537988749312756711033825, 7.05236921534460662134368459184, 7.71797687284700530203178504871, 8.350359779199986946535701074361, 9.285249648750112101353825405682

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