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Properties

Label 2-2576-2576.195-c0-0-1
Degree $2$
Conductor $2576$
Sign $0.735 - 0.677i$
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  + (−0.989 + 0.142i)2-s + (0.959 − 0.281i)4-s + (0.755 − 0.654i)7-s + (−0.909 + 0.415i)8-s + (−0.281 + 0.959i)9-s + (−1.17 + 0.254i)11-s + (−0.654 + 0.755i)14-s + (0.841 − 0.540i)16-s + (0.142 − 0.989i)18-s + (1.12 − 0.418i)22-s + (0.959 − 0.281i)23-s + (0.540 + 0.841i)25-s + (0.540 − 0.841i)28-s + (−0.148 + 0.398i)29-s + (−0.755 + 0.654i)32-s + ⋯
L(s)  = 1  + (−0.989 + 0.142i)2-s + (0.959 − 0.281i)4-s + (0.755 − 0.654i)7-s + (−0.909 + 0.415i)8-s + (−0.281 + 0.959i)9-s + (−1.17 + 0.254i)11-s + (−0.654 + 0.755i)14-s + (0.841 − 0.540i)16-s + (0.142 − 0.989i)18-s + (1.12 − 0.418i)22-s + (0.959 − 0.281i)23-s + (0.540 + 0.841i)25-s + (0.540 − 0.841i)28-s + (−0.148 + 0.398i)29-s + (−0.755 + 0.654i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $0.735 - 0.677i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2576} (195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2576,\ (\ :0),\ 0.735 - 0.677i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7664896636\)
\(L(\frac12)\) \(\approx\) \(0.7664896636\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.989 - 0.142i)T \)
7 \( 1 + (-0.755 + 0.654i)T \)
23 \( 1 + (-0.959 + 0.281i)T \)
good3 \( 1 + (0.281 - 0.959i)T^{2} \)
5 \( 1 + (-0.540 - 0.841i)T^{2} \)
11 \( 1 + (1.17 - 0.254i)T + (0.909 - 0.415i)T^{2} \)
13 \( 1 + (-0.989 + 0.142i)T^{2} \)
17 \( 1 + (-0.654 + 0.755i)T^{2} \)
19 \( 1 + (-0.755 + 0.654i)T^{2} \)
29 \( 1 + (0.148 - 0.398i)T + (-0.755 - 0.654i)T^{2} \)
31 \( 1 + (-0.959 + 0.281i)T^{2} \)
37 \( 1 + (-0.956 - 1.75i)T + (-0.540 + 0.841i)T^{2} \)
41 \( 1 + (0.841 - 0.540i)T^{2} \)
43 \( 1 + (-1.05 - 1.40i)T + (-0.281 + 0.959i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.0303 + 0.424i)T + (-0.989 - 0.142i)T^{2} \)
59 \( 1 + (-0.989 + 0.142i)T^{2} \)
61 \( 1 + (-0.281 - 0.959i)T^{2} \)
67 \( 1 + (0.682 + 0.148i)T + (0.909 + 0.415i)T^{2} \)
71 \( 1 + (-1.66 + 1.07i)T + (0.415 - 0.909i)T^{2} \)
73 \( 1 + (-0.654 - 0.755i)T^{2} \)
79 \( 1 + (-0.708 + 0.817i)T + (-0.142 - 0.989i)T^{2} \)
83 \( 1 + (0.540 - 0.841i)T^{2} \)
89 \( 1 + (0.959 + 0.281i)T^{2} \)
97 \( 1 + (0.841 - 0.540i)T^{2} \)
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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

−9.078158810022803552437211885007, −8.224486489806446100559715908647, −7.76939120227858833337757507762, −7.22610076720513121630377657038, −6.26405896372120818844069861775, −5.15333824304382871504913608183, −4.77100652434922851141200744737, −3.15073414246167248076300019567, −2.30604730181341867221077789727, −1.19533075862526364022710660920, 0.791199059974743155892091228911, 2.25544713224958196728360523873, 2.87671441262190278799329571216, 4.04429863897909677994168520309, 5.35820481905064184344482223931, 5.89826697532957247964755327380, 6.88563847017648949439856615102, 7.67121687491892308295880957870, 8.327431564883915645223277279044, 8.989798750740442292715682834089

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