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Properties

Label 2-2496-2496.1637-c0-0-3
Degree $2$
Conductor $2496$
Sign $-0.995 - 0.0980i$
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  + (−0.290 − 0.956i)2-s + (0.555 − 0.831i)3-s + (−0.831 + 0.555i)4-s + (0.373 − 1.87i)5-s + (−0.956 − 0.290i)6-s + (0.773 + 0.634i)8-s + (−0.382 − 0.923i)9-s + (−1.90 + 0.187i)10-s + (1.05 − 0.704i)11-s + i·12-s + (0.195 + 0.980i)13-s + (−1.35 − 1.35i)15-s + (0.382 − 0.923i)16-s + (−0.773 + 0.634i)18-s + (0.732 + 1.76i)20-s + ⋯
L(s)  = 1  + (−0.290 − 0.956i)2-s + (0.555 − 0.831i)3-s + (−0.831 + 0.555i)4-s + (0.373 − 1.87i)5-s + (−0.956 − 0.290i)6-s + (0.773 + 0.634i)8-s + (−0.382 − 0.923i)9-s + (−1.90 + 0.187i)10-s + (1.05 − 0.704i)11-s + i·12-s + (0.195 + 0.980i)13-s + (−1.35 − 1.35i)15-s + (0.382 − 0.923i)16-s + (−0.773 + 0.634i)18-s + (0.732 + 1.76i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $-0.995 - 0.0980i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (1637, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :0),\ -0.995 - 0.0980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.282537470\)
\(L(\frac12)\) \(\approx\) \(1.282537470\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.290 + 0.956i)T \)
3 \( 1 + (-0.555 + 0.831i)T \)
13 \( 1 + (-0.195 - 0.980i)T \)
good5 \( 1 + (-0.373 + 1.87i)T + (-0.923 - 0.382i)T^{2} \)
7 \( 1 + (0.707 + 0.707i)T^{2} \)
11 \( 1 + (-1.05 + 0.704i)T + (0.382 - 0.923i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (-0.923 + 0.382i)T^{2} \)
23 \( 1 + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (-0.382 - 0.923i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.923 - 0.382i)T^{2} \)
41 \( 1 + (-1.62 + 0.674i)T + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (-1.08 - 1.63i)T + (-0.382 + 0.923i)T^{2} \)
47 \( 1 + (0.138 - 0.138i)T - iT^{2} \)
53 \( 1 + (-0.382 + 0.923i)T^{2} \)
59 \( 1 + (0.113 - 0.569i)T + (-0.923 - 0.382i)T^{2} \)
61 \( 1 + (1.02 - 1.53i)T + (-0.382 - 0.923i)T^{2} \)
67 \( 1 + (0.382 + 0.923i)T^{2} \)
71 \( 1 + (0.591 - 1.42i)T + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (-1.38 - 1.38i)T + iT^{2} \)
83 \( 1 + (0.924 - 0.183i)T + (0.923 - 0.382i)T^{2} \)
89 \( 1 + (0.181 + 0.0750i)T + (0.707 + 0.707i)T^{2} \)
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

−8.996909627475874616165118968728, −8.288316472682817700496503328875, −7.53942917173911089137535278385, −6.33068794201133401298502948604, −5.55137021090149229014805543999, −4.36999677516594779823720613158, −3.91769415583134252754124001802, −2.58700866387824112261683939789, −1.51751640919983830987225901512, −1.00354342356160520564766820760, 2.00119381698266075162230870283, 3.14626119265158391171429642869, 3.84899145667485785430969798673, 4.81221885248677512845428936877, 5.93477370370968085207523839823, 6.36057305124185725868185828062, 7.42140251338559202801986415564, 7.67966879020268689961152383940, 8.815229868220901242807570782489, 9.561180503479644930843621093716

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