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Properties

Label 2-2576-2576.1259-c0-0-1
Degree $2$
Conductor $2576$
Sign $-0.394 + 0.918i$
Analytic cond. $1.28559$
Root an. cond. $1.13383$
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  + (−0.909 + 0.415i)2-s + (0.654 − 0.755i)4-s + (−0.989 − 0.142i)7-s + (−0.281 + 0.959i)8-s + (−0.540 + 0.841i)9-s + (0.613 − 1.64i)11-s + (0.959 − 0.281i)14-s + (−0.142 − 0.989i)16-s + (0.142 − 0.989i)18-s + (0.125 + 1.75i)22-s + (−0.841 + 0.540i)23-s + (−0.909 − 0.415i)25-s + (−0.755 + 0.654i)28-s + (−0.0498 − 0.697i)29-s + (0.540 + 0.841i)32-s + ⋯
L(s)  = 1  + (−0.909 + 0.415i)2-s + (0.654 − 0.755i)4-s + (−0.989 − 0.142i)7-s + (−0.281 + 0.959i)8-s + (−0.540 + 0.841i)9-s + (0.613 − 1.64i)11-s + (0.959 − 0.281i)14-s + (−0.142 − 0.989i)16-s + (0.142 − 0.989i)18-s + (0.125 + 1.75i)22-s + (−0.841 + 0.540i)23-s + (−0.909 − 0.415i)25-s + (−0.755 + 0.654i)28-s + (−0.0498 − 0.697i)29-s + (0.540 + 0.841i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $-0.394 + 0.918i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2576} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2576,\ (\ :0),\ -0.394 + 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3002163111\)
\(L(\frac12)\) \(\approx\) \(0.3002163111\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.909 - 0.415i)T \)
7 \( 1 + (0.989 + 0.142i)T \)
23 \( 1 + (0.841 - 0.540i)T \)
good3 \( 1 + (0.540 - 0.841i)T^{2} \)
5 \( 1 + (0.909 + 0.415i)T^{2} \)
11 \( 1 + (-0.613 + 1.64i)T + (-0.755 - 0.654i)T^{2} \)
13 \( 1 + (-0.281 - 0.959i)T^{2} \)
17 \( 1 + (-0.142 - 0.989i)T^{2} \)
19 \( 1 + (-0.989 - 0.142i)T^{2} \)
29 \( 1 + (0.0498 + 0.697i)T + (-0.989 + 0.142i)T^{2} \)
31 \( 1 + (0.841 - 0.540i)T^{2} \)
37 \( 1 + (1.17 - 0.254i)T + (0.909 - 0.415i)T^{2} \)
41 \( 1 + (0.415 - 0.909i)T^{2} \)
43 \( 1 + (0.936 + 1.71i)T + (-0.540 + 0.841i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.418 - 0.559i)T + (-0.281 + 0.959i)T^{2} \)
59 \( 1 + (-0.281 - 0.959i)T^{2} \)
61 \( 1 + (-0.540 - 0.841i)T^{2} \)
67 \( 1 + (0.697 + 1.86i)T + (-0.755 + 0.654i)T^{2} \)
71 \( 1 + (0.234 - 0.512i)T + (-0.654 - 0.755i)T^{2} \)
73 \( 1 + (-0.142 + 0.989i)T^{2} \)
79 \( 1 + (0.258 + 1.80i)T + (-0.959 + 0.281i)T^{2} \)
83 \( 1 + (-0.909 + 0.415i)T^{2} \)
89 \( 1 + (-0.841 - 0.540i)T^{2} \)
97 \( 1 + (0.415 - 0.909i)T^{2} \)
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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

−8.782701128546352926154758117712, −8.173161795421679394116306459521, −7.45633719112205124469099064838, −6.47156218200503712104937343607, −5.96120500110258002123930911991, −5.31852254749112709703643520933, −3.84268259181640345713120322185, −2.97280984113438558372980239786, −1.84648083807085012725698807368, −0.25426871897691598686108654569, 1.52950464894460789732932610960, 2.57633506375725052120385458658, 3.52639004488207027431170784182, 4.23251071425178340163533635948, 5.65112891379669489460582727880, 6.64674244132815811313403206253, 6.90257490369359608790654363250, 7.906706456850868917605978346758, 8.787448444653702581562343485501, 9.415359380311131972881255529419

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