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Properties

Label 2-2576-2576.13-c0-0-1
Degree $2$
Conductor $2576$
Sign $-0.0239 + 0.999i$
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 + (−0.139 − 1.94i)11-s + (0.959 − 0.281i)14-s + (−0.654 − 0.755i)16-s + (0.841 + 0.540i)18-s + (−1.56 + 1.17i)22-s + (0.415 − 0.909i)23-s + (−0.755 + 0.654i)25-s + (−0.755 − 0.654i)28-s + (1.19 − 1.59i)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 + (−0.139 − 1.94i)11-s + (0.959 − 0.281i)14-s + (−0.654 − 0.755i)16-s + (0.841 + 0.540i)18-s + (−1.56 + 1.17i)22-s + (0.415 − 0.909i)23-s + (−0.755 + 0.654i)25-s + (−0.755 − 0.654i)28-s + (1.19 − 1.59i)29-s + (−0.281 + 0.959i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0239 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0239 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $-0.0239 + 0.999i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2576} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2576,\ (\ :0),\ -0.0239 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6984873201\)
\(L(\frac12)\) \(\approx\) \(0.6984873201\)
\(L(1)\) 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 \( 1 + (0.139 + 1.94i)T + (-0.989 + 0.142i)T^{2} \)
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 \( 1 + (-1.19 + 1.59i)T + (-0.281 - 0.959i)T^{2} \)
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 \( 1 + (-1.83 + 0.398i)T + (0.909 - 0.415i)T^{2} \)
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 \( 1 + (-0.0855 + 1.19i)T + (-0.989 - 0.142i)T^{2} \)
71 \( 1 + (-1.27 + 1.10i)T + (0.142 - 0.989i)T^{2} \)
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

−9.009107164149861369104264479730, −8.120359007111423456880685356854, −7.988366782870324647176091852056, −6.39422110757435669197614484088, −5.83942809032379176473533921386, −4.92617810595038980338491827077, −3.72307219649457010037233399390, −2.85813080483220442555636919067, −2.35975424483627304698497840717, −0.63816305107060036858668381661, 1.15114945678127293942038818241, 2.49672315262157766694916825226, 3.89468230059604468566319863938, 4.65699680194807396348718999031, 5.48024691597111560271241662520, 6.43936331052024389804229568453, 7.05834808315217750937752461483, 7.62241782547264145941047305361, 8.388570140838193654257548747560, 9.416140101112189968672768303437

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