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Properties

Label 2-2496-2496.1013-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  + (0.773 − 0.634i)2-s + (−0.980 + 0.195i)3-s + (0.195 − 0.980i)4-s + (−0.704 − 1.05i)5-s + (−0.634 + 0.773i)6-s + (−0.471 − 0.881i)8-s + (0.923 − 0.382i)9-s + (−1.21 − 0.368i)10-s + (0.344 − 1.72i)11-s + i·12-s + (−0.555 + 0.831i)13-s + (0.897 + 0.897i)15-s + (−0.923 − 0.382i)16-s + (0.471 − 0.881i)18-s + (−1.17 + 0.485i)20-s + ⋯
L(s)  = 1  + (0.773 − 0.634i)2-s + (−0.980 + 0.195i)3-s + (0.195 − 0.980i)4-s + (−0.704 − 1.05i)5-s + (−0.634 + 0.773i)6-s + (−0.471 − 0.881i)8-s + (0.923 − 0.382i)9-s + (−1.21 − 0.368i)10-s + (0.344 − 1.72i)11-s + i·12-s + (−0.555 + 0.831i)13-s + (0.897 + 0.897i)15-s + (−0.923 − 0.382i)16-s + (0.471 − 0.881i)18-s + (−1.17 + 0.485i)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: $\chi_{2496} (1013, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :0),\ -0.995 - 0.0980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8764614477\)
\(L(\frac12)\) \(\approx\) \(0.8764614477\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.773 + 0.634i)T \)
3 \( 1 + (0.980 - 0.195i)T \)
13 \( 1 + (0.555 - 0.831i)T \)
good5 \( 1 + (0.704 + 1.05i)T + (-0.382 + 0.923i)T^{2} \)
7 \( 1 + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + (-0.344 + 1.72i)T + (-0.923 - 0.382i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (-0.382 - 0.923i)T^{2} \)
23 \( 1 + (0.707 - 0.707i)T^{2} \)
29 \( 1 + (0.923 - 0.382i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.382 + 0.923i)T^{2} \)
41 \( 1 + (-0.0750 - 0.181i)T + (-0.707 + 0.707i)T^{2} \)
43 \( 1 + (1.63 + 0.324i)T + (0.923 + 0.382i)T^{2} \)
47 \( 1 + (0.410 - 0.410i)T - iT^{2} \)
53 \( 1 + (0.923 + 0.382i)T^{2} \)
59 \( 1 + (0.858 + 1.28i)T + (-0.382 + 0.923i)T^{2} \)
61 \( 1 + (-0.750 + 0.149i)T + (0.923 - 0.382i)T^{2} \)
67 \( 1 + (-0.923 + 0.382i)T^{2} \)
71 \( 1 + (0.871 + 0.360i)T + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (1.17 + 1.17i)T + iT^{2} \)
83 \( 1 + (-1.65 - 1.10i)T + (0.382 + 0.923i)T^{2} \)
89 \( 1 + (-0.222 + 0.536i)T + (-0.707 - 0.707i)T^{2} \)
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

−8.940768858200949626640364485106, −8.003442304909304507048887831736, −6.83151710693256538646146386190, −6.16717186771984254828651018204, −5.36473491779769090361038515630, −4.70463486155057457838906070728, −4.04192167493338227821495543669, −3.21911976996015960379657072526, −1.59289662263309545600580239203, −0.51198847737137961292475501440, 2.04845206596814234698292923110, 3.16265183741172885092032932846, 4.15586515292277887890818742525, 4.81701614397770816783183653803, 5.61576608619020510552449380306, 6.56303859640052445368569984241, 7.18496427303500522180888462524, 7.40650251397706195308296277117, 8.345769889847921167411866554268, 9.701462937439132119054116844371

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