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Properties

Label 2-2576-2576.1203-c0-0-1
Degree $2$
Conductor $2576$
Sign $-0.998 - 0.0606i$
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  + (0.540 − 0.841i)2-s + (−0.415 − 0.909i)4-s + (−0.281 − 0.959i)7-s + (−0.989 − 0.142i)8-s + (−0.909 − 0.415i)9-s + (0.424 + 0.0303i)11-s + (−0.959 − 0.281i)14-s + (−0.654 + 0.755i)16-s + (−0.841 + 0.540i)18-s + (0.254 − 0.340i)22-s + (−0.415 − 0.909i)23-s + (−0.755 − 0.654i)25-s + (−0.755 + 0.654i)28-s + (−0.114 + 0.0855i)29-s + (0.281 + 0.959i)32-s + ⋯
L(s)  = 1  + (0.540 − 0.841i)2-s + (−0.415 − 0.909i)4-s + (−0.281 − 0.959i)7-s + (−0.989 − 0.142i)8-s + (−0.909 − 0.415i)9-s + (0.424 + 0.0303i)11-s + (−0.959 − 0.281i)14-s + (−0.654 + 0.755i)16-s + (−0.841 + 0.540i)18-s + (0.254 − 0.340i)22-s + (−0.415 − 0.909i)23-s + (−0.755 − 0.654i)25-s + (−0.755 + 0.654i)28-s + (−0.114 + 0.0855i)29-s + (0.281 + 0.959i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $-0.998 - 0.0606i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2576} (1203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2576,\ (\ :0),\ -0.998 - 0.0606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.005755534\)
\(L(\frac12)\) \(\approx\) \(1.005755534\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.540 + 0.841i)T \)
7 \( 1 + (0.281 + 0.959i)T \)
23 \( 1 + (0.415 + 0.909i)T \)
good3 \( 1 + (0.909 + 0.415i)T^{2} \)
5 \( 1 + (0.755 + 0.654i)T^{2} \)
11 \( 1 + (-0.424 - 0.0303i)T + (0.989 + 0.142i)T^{2} \)
13 \( 1 + (0.540 - 0.841i)T^{2} \)
17 \( 1 + (-0.959 - 0.281i)T^{2} \)
19 \( 1 + (0.281 + 0.959i)T^{2} \)
29 \( 1 + (0.114 - 0.0855i)T + (0.281 - 0.959i)T^{2} \)
31 \( 1 + (0.415 + 0.909i)T^{2} \)
37 \( 1 + (0.898 - 0.334i)T + (0.755 - 0.654i)T^{2} \)
41 \( 1 + (-0.654 + 0.755i)T^{2} \)
43 \( 1 + (-0.148 + 0.682i)T + (-0.909 - 0.415i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.125 + 0.0683i)T + (0.540 + 0.841i)T^{2} \)
59 \( 1 + (0.540 - 0.841i)T^{2} \)
61 \( 1 + (-0.909 + 0.415i)T^{2} \)
67 \( 1 + (-1.59 + 0.114i)T + (0.989 - 0.142i)T^{2} \)
71 \( 1 + (-0.708 + 0.817i)T + (-0.142 - 0.989i)T^{2} \)
73 \( 1 + (-0.959 + 0.281i)T^{2} \)
79 \( 1 + (1.45 + 0.425i)T + (0.841 + 0.540i)T^{2} \)
83 \( 1 + (-0.755 + 0.654i)T^{2} \)
89 \( 1 + (-0.415 + 0.909i)T^{2} \)
97 \( 1 + (-0.654 + 0.755i)T^{2} \)
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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

−8.827074245740046270943377357205, −8.104708170130495229116516291322, −6.89388845681847386757666601415, −6.29283670624010263989851327244, −5.49069420877809681912305037371, −4.47304444581600137111584681899, −3.78822226240639418945956644995, −3.03086467469483516285667191980, −1.92253296821525092810117574899, −0.52243650857941099366671621674, 2.11240688116067368566298255296, 3.11946924229025663660271940744, 3.88096809461185319228201124200, 5.06356116865695467564937814931, 5.63555785844933805001006846375, 6.19283676434016236738922715548, 7.11018685990563930513767203237, 7.939373299291906312530385810223, 8.568570593438481412986799557245, 9.221174338615337416840058134328

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