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Properties

Label 2-2576-2576.1917-c0-0-1
Degree $2$
Conductor $2576$
Sign $-0.141 + 0.989i$
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 + (0.540 − 0.841i)7-s + (−0.909 + 0.415i)8-s + (−0.755 − 0.654i)9-s + (−0.559 − 0.418i)11-s + (−0.415 + 0.909i)14-s + (0.841 − 0.540i)16-s + (0.841 + 0.540i)18-s + (0.613 + 0.334i)22-s + (0.654 + 0.755i)23-s + (0.989 + 0.142i)25-s + (0.281 − 0.959i)28-s + (−1.40 − 0.767i)29-s + (−0.755 + 0.654i)32-s + ⋯
L(s)  = 1  + (−0.989 + 0.142i)2-s + (0.959 − 0.281i)4-s + (0.540 − 0.841i)7-s + (−0.909 + 0.415i)8-s + (−0.755 − 0.654i)9-s + (−0.559 − 0.418i)11-s + (−0.415 + 0.909i)14-s + (0.841 − 0.540i)16-s + (0.841 + 0.540i)18-s + (0.613 + 0.334i)22-s + (0.654 + 0.755i)23-s + (0.989 + 0.142i)25-s + (0.281 − 0.959i)28-s + (−1.40 − 0.767i)29-s + (−0.755 + 0.654i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.141 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.141 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $-0.141 + 0.989i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2576} (1917, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2576,\ (\ :0),\ -0.141 + 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5964886828\)
\(L(\frac12)\) \(\approx\) \(0.5964886828\)
\(L(1)\) 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 \( 1 + (-0.540 + 0.841i)T \)
23 \( 1 + (-0.654 - 0.755i)T \)
good3 \( 1 + (0.755 + 0.654i)T^{2} \)
5 \( 1 + (-0.989 - 0.142i)T^{2} \)
11 \( 1 + (0.559 + 0.418i)T + (0.281 + 0.959i)T^{2} \)
13 \( 1 + (-0.909 - 0.415i)T^{2} \)
17 \( 1 + (-0.841 + 0.540i)T^{2} \)
19 \( 1 + (0.540 - 0.841i)T^{2} \)
29 \( 1 + (1.40 + 0.767i)T + (0.540 + 0.841i)T^{2} \)
31 \( 1 + (0.654 + 0.755i)T^{2} \)
37 \( 1 + (0.139 + 1.94i)T + (-0.989 + 0.142i)T^{2} \)
41 \( 1 + (-0.142 + 0.989i)T^{2} \)
43 \( 1 + (1.86 + 0.697i)T + (0.755 + 0.654i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-1.17 + 0.254i)T + (0.909 - 0.415i)T^{2} \)
59 \( 1 + (0.909 + 0.415i)T^{2} \)
61 \( 1 + (-0.755 + 0.654i)T^{2} \)
67 \( 1 + (-0.767 + 0.574i)T + (0.281 - 0.959i)T^{2} \)
71 \( 1 + (0.822 + 0.118i)T + (0.959 + 0.281i)T^{2} \)
73 \( 1 + (0.841 + 0.540i)T^{2} \)
79 \( 1 + (1.66 - 1.07i)T + (0.415 - 0.909i)T^{2} \)
83 \( 1 + (-0.989 + 0.142i)T^{2} \)
89 \( 1 + (-0.654 + 0.755i)T^{2} \)
97 \( 1 + (0.142 - 0.989i)T^{2} \)
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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

−8.776999271220007664706600167584, −8.231290683122009771736927814877, −7.34641632138058318221847747528, −6.91166009074909134339365551059, −5.78152918219002649085498666099, −5.27036393804636287260774381976, −3.84851089962374438105128411742, −3.00905894066157234036639459028, −1.82109650956773504090769979272, −0.52871186840828446268047423759, 1.55161522588317576324276805218, 2.51716145558840193956414880682, 3.16283274981276010099130928596, 4.79105356949548595570638962658, 5.39076633747494063912307005488, 6.35578897452084162501664543116, 7.19388263941456617799277440825, 7.990254074230280269605872669888, 8.611643795795871828338713856858, 9.004792648713225408469240659996

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