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Properties

Label 2-2576-2576.1637-c0-0-0
Degree $2$
Conductor $2576$
Sign $0.835 - 0.549i$
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 + (0.898 − 0.334i)11-s + (0.959 + 0.281i)14-s + (−0.142 + 0.989i)16-s + (−0.142 − 0.989i)18-s + (0.956 + 0.0683i)22-s + (−0.841 − 0.540i)23-s + (−0.909 + 0.415i)25-s + (0.755 + 0.654i)28-s + (1.86 + 0.133i)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 + (0.898 − 0.334i)11-s + (0.959 + 0.281i)14-s + (−0.142 + 0.989i)16-s + (−0.142 − 0.989i)18-s + (0.956 + 0.0683i)22-s + (−0.841 − 0.540i)23-s + (−0.909 + 0.415i)25-s + (0.755 + 0.654i)28-s + (1.86 + 0.133i)29-s + (−0.540 + 0.841i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $0.835 - 0.549i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2576} (1637, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2576,\ (\ :0),\ 0.835 - 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.331072135\)
\(L(\frac12)\) \(\approx\) \(2.331072135\)
\(L(1)\) 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 \( 1 + (-0.898 + 0.334i)T + (0.755 - 0.654i)T^{2} \)
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 \( 1 + (-1.86 - 0.133i)T + (0.989 + 0.142i)T^{2} \)
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 \( 1 + (0.373 + 0.203i)T + (0.540 + 0.841i)T^{2} \)
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 \( 1 + (-0.133 - 0.0498i)T + (0.755 + 0.654i)T^{2} \)
71 \( 1 + (1.74 - 0.797i)T + (0.654 - 0.755i)T^{2} \)
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

−8.704880466444809134130423523449, −8.457552049692128961097847846674, −7.48913218081985709796387351084, −6.66686939074427816812127545849, −6.05134199515572285398538835080, −5.27415888183065066900704951991, −4.33238969116068987693408342786, −3.72451357461931201072457034608, −2.71373748296059653506859003957, −1.48707813190165640912902884507, 1.52957552484683635548837714516, 2.24407418164890665112996362510, 3.35277689380182084305259184912, 4.39184249894459039662945121607, 4.88952852949723212795535388199, 5.77163126976237374100422402071, 6.46661806663138913820883551781, 7.45971871194161911297909143587, 8.168956491106167364659304858865, 8.983407723036528178706205923108

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