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Properties

Label 2-2496-2496.701-c0-0-0
Degree $2$
Conductor $2496$
Sign $-0.0980 - 0.995i$
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  + (−0.0980 − 0.995i)2-s + (−0.195 + 0.980i)3-s + (−0.980 + 0.195i)4-s + (1.65 + 1.10i)5-s + (0.995 + 0.0980i)6-s + (0.290 + 0.956i)8-s + (−0.923 − 0.382i)9-s + (0.938 − 1.75i)10-s + (−1.87 + 0.373i)11-s − i·12-s + (−0.831 + 0.555i)13-s + (−1.40 + 1.40i)15-s + (0.923 − 0.382i)16-s + (−0.290 + 0.956i)18-s + (−1.83 − 0.761i)20-s + ⋯
L(s)  = 1  + (−0.0980 − 0.995i)2-s + (−0.195 + 0.980i)3-s + (−0.980 + 0.195i)4-s + (1.65 + 1.10i)5-s + (0.995 + 0.0980i)6-s + (0.290 + 0.956i)8-s + (−0.923 − 0.382i)9-s + (0.938 − 1.75i)10-s + (−1.87 + 0.373i)11-s − i·12-s + (−0.831 + 0.555i)13-s + (−1.40 + 1.40i)15-s + (0.923 − 0.382i)16-s + (−0.290 + 0.956i)18-s + (−1.83 − 0.761i)20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $-0.0980 - 0.995i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :0),\ -0.0980 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8727458598\)
\(L(\frac12)\) \(\approx\) \(0.8727458598\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0980 + 0.995i)T \)
3 \( 1 + (0.195 - 0.980i)T \)
13 \( 1 + (0.831 - 0.555i)T \)
good5 \( 1 + (-1.65 - 1.10i)T + (0.382 + 0.923i)T^{2} \)
7 \( 1 + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (1.87 - 0.373i)T + (0.923 - 0.382i)T^{2} \)
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 \( 1 + (0.485 - 1.17i)T + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + (-0.216 - 1.08i)T + (-0.923 + 0.382i)T^{2} \)
47 \( 1 + (1.24 + 1.24i)T + iT^{2} \)
53 \( 1 + (-0.923 + 0.382i)T^{2} \)
59 \( 1 + (-0.162 - 0.108i)T + (0.382 + 0.923i)T^{2} \)
61 \( 1 + (0.149 - 0.750i)T + (-0.923 - 0.382i)T^{2} \)
67 \( 1 + (0.923 + 0.382i)T^{2} \)
71 \( 1 + (0.536 - 0.222i)T + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (-0.785 + 0.785i)T - iT^{2} \)
83 \( 1 + (-0.858 - 1.28i)T + (-0.382 + 0.923i)T^{2} \)
89 \( 1 + (0.674 + 1.62i)T + (-0.707 + 0.707i)T^{2} \)
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

−9.782291054659788257418019236606, −8.986790320370815365803641756612, −7.975602651399804614669829686342, −6.95070323628793773021184552536, −5.90098936508976654980808761085, −5.20864275836020003136719776705, −4.68011552572796373276717858820, −3.28241692984571135391126436852, −2.65114351639444037280194367353, −2.00356406990138287357148340482, 0.56978382850206409968642802564, 1.90663334105115762166850032525, 2.83934345165553941304238742236, 4.77306622373163372778328828311, 5.34229951427232995986069011906, 5.68717023012601681316389141228, 6.47660681849678934842272225107, 7.44530144622454399246317039925, 8.079697229448953276824478250087, 8.679880734188659239250659081631

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