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Properties

Label 2-2496-2496.1637-c0-0-2
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  + (0.956 − 0.290i)2-s + (−0.555 + 0.831i)3-s + (0.831 − 0.555i)4-s + (−0.113 + 0.569i)5-s + (−0.290 + 0.956i)6-s + (0.634 − 0.773i)8-s + (−0.382 − 0.923i)9-s + (0.0569 + 0.577i)10-s + (1.28 − 0.858i)11-s + i·12-s + (−0.195 − 0.980i)13-s + (−0.410 − 0.410i)15-s + (0.382 − 0.923i)16-s + (−0.634 − 0.773i)18-s + (0.222 + 0.536i)20-s + ⋯
L(s)  = 1  + (0.956 − 0.290i)2-s + (−0.555 + 0.831i)3-s + (0.831 − 0.555i)4-s + (−0.113 + 0.569i)5-s + (−0.290 + 0.956i)6-s + (0.634 − 0.773i)8-s + (−0.382 − 0.923i)9-s + (0.0569 + 0.577i)10-s + (1.28 − 0.858i)11-s + i·12-s + (−0.195 − 0.980i)13-s + (−0.410 − 0.410i)15-s + (0.382 − 0.923i)16-s + (−0.634 − 0.773i)18-s + (0.222 + 0.536i)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

\(L(\frac{1}{2})\) \(\approx\) \(1.919620364\)
\(L(\frac12)\) \(\approx\) \(1.919620364\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.956 + 0.290i)T \)
3 \( 1 + (0.555 - 0.831i)T \)
13 \( 1 + (0.195 + 0.980i)T \)
good5 \( 1 + (0.113 - 0.569i)T + (-0.923 - 0.382i)T^{2} \)
7 \( 1 + (0.707 + 0.707i)T^{2} \)
11 \( 1 + (-1.28 + 0.858i)T + (0.382 - 0.923i)T^{2} \)
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 \( 1 + (0.871 - 0.360i)T + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (-1.08 - 1.63i)T + (-0.382 + 0.923i)T^{2} \)
47 \( 1 + (1.40 - 1.40i)T - iT^{2} \)
53 \( 1 + (-0.382 + 0.923i)T^{2} \)
59 \( 1 + (0.373 - 1.87i)T + (-0.923 - 0.382i)T^{2} \)
61 \( 1 + (-1.02 + 1.53i)T + (-0.382 - 0.923i)T^{2} \)
67 \( 1 + (0.382 + 0.923i)T^{2} \)
71 \( 1 + (0.485 - 1.17i)T + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (1.38 + 1.38i)T + iT^{2} \)
83 \( 1 + (-1.72 + 0.344i)T + (0.923 - 0.382i)T^{2} \)
89 \( 1 + (1.83 + 0.761i)T + (0.707 + 0.707i)T^{2} \)
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

−9.376613704356095846643213529487, −8.384025824187219862769457505843, −7.27150318684304744564746606757, −6.37158695925103548607577931223, −6.00622199586767306113610951881, −5.05878274745413284934498693187, −4.31079948016353595551429766036, −3.36800674126181484505681771003, −2.96512919334527110449061882688, −1.19025519598771751878518753415, 1.48191395448397869626170407210, 2.24900566509235436718281643901, 3.68738686575504630726234828732, 4.54450879919775018856081652791, 5.12715591075459257822732384763, 6.08977213893866322426153280822, 6.92244128626406031267153310025, 7.06246830390961156696523773204, 8.212050330124887109295082252886, 8.868791141950290252630293775839

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