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Properties

Label 2-2496-2496.77-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  + (−0.471 − 0.881i)2-s + (−0.831 + 0.555i)3-s + (−0.555 + 0.831i)4-s + (−1.72 + 0.344i)5-s + (0.881 + 0.471i)6-s + (0.995 + 0.0980i)8-s + (0.382 − 0.923i)9-s + (1.11 + 1.36i)10-s + (−0.108 + 0.162i)11-s − i·12-s + (0.980 + 0.195i)13-s + (1.24 − 1.24i)15-s + (−0.382 − 0.923i)16-s + (−0.995 + 0.0980i)18-s + (0.674 − 1.62i)20-s + ⋯
L(s)  = 1  + (−0.471 − 0.881i)2-s + (−0.831 + 0.555i)3-s + (−0.555 + 0.831i)4-s + (−1.72 + 0.344i)5-s + (0.881 + 0.471i)6-s + (0.995 + 0.0980i)8-s + (0.382 − 0.923i)9-s + (1.11 + 1.36i)10-s + (−0.108 + 0.162i)11-s − i·12-s + (0.980 + 0.195i)13-s + (1.24 − 1.24i)15-s + (−0.382 − 0.923i)16-s + (−0.995 + 0.0980i)18-s + (0.674 − 1.62i)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: $\chi_{2496} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :0),\ -0.0980 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2659091981\)
\(L(\frac12)\) \(\approx\) \(0.2659091981\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.471 + 0.881i)T \)
3 \( 1 + (0.831 - 0.555i)T \)
13 \( 1 + (-0.980 - 0.195i)T \)
good5 \( 1 + (1.72 - 0.344i)T + (0.923 - 0.382i)T^{2} \)
7 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.108 - 0.162i)T + (-0.382 - 0.923i)T^{2} \)
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 \( 1 + (-0.536 - 0.222i)T + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (-0.324 - 0.216i)T + (0.382 + 0.923i)T^{2} \)
47 \( 1 + (1.09 + 1.09i)T + iT^{2} \)
53 \( 1 + (0.382 + 0.923i)T^{2} \)
59 \( 1 + (0.924 - 0.183i)T + (0.923 - 0.382i)T^{2} \)
61 \( 1 + (1.53 - 1.02i)T + (0.382 - 0.923i)T^{2} \)
67 \( 1 + (-0.382 + 0.923i)T^{2} \)
71 \( 1 + (-0.761 - 1.83i)T + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (0.275 - 0.275i)T - iT^{2} \)
83 \( 1 + (0.373 - 1.87i)T + (-0.923 - 0.382i)T^{2} \)
89 \( 1 + (-1.42 + 0.591i)T + (0.707 - 0.707i)T^{2} \)
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

−9.392306455193316000331527677319, −8.617770860050407655606497469873, −7.922583976881256315588940589668, −7.18229587729467657802515282489, −6.33668040844271933766059262995, −5.05364005716704274167607296360, −4.22424286204676531371942055114, −3.76009835798916428817294301529, −2.92181283095868249620818154914, −1.15258419804464861581908681857, 0.29617908706340276564080698536, 1.45776270866997950108013675034, 3.42010647005122127951423262173, 4.43186894267887498358314053488, 5.01300215354758411114108766862, 6.04149279134545352737000582685, 6.62952838870393028636565816863, 7.61496414265307545640756963918, 7.86614763824635975744495654515, 8.573274284387035547990493789044

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