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Properties

Label 2-2576-2576.1973-c0-0-0
Degree $2$
Conductor $2576$
Sign $-0.490 + 0.871i$
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  + (−0.281 − 0.959i)2-s + (−0.841 + 0.540i)4-s + (−0.909 + 0.415i)7-s + (0.755 + 0.654i)8-s + (−0.989 + 0.142i)9-s + (0.125 + 0.0683i)11-s + (0.654 + 0.755i)14-s + (0.415 − 0.909i)16-s + (0.415 + 0.909i)18-s + (0.0303 − 0.139i)22-s + (0.142 − 0.989i)23-s + (0.281 − 0.959i)25-s + (0.540 − 0.841i)28-s + (0.373 − 1.71i)29-s + (−0.989 − 0.142i)32-s + ⋯
L(s)  = 1  + (−0.281 − 0.959i)2-s + (−0.841 + 0.540i)4-s + (−0.909 + 0.415i)7-s + (0.755 + 0.654i)8-s + (−0.989 + 0.142i)9-s + (0.125 + 0.0683i)11-s + (0.654 + 0.755i)14-s + (0.415 − 0.909i)16-s + (0.415 + 0.909i)18-s + (0.0303 − 0.139i)22-s + (0.142 − 0.989i)23-s + (0.281 − 0.959i)25-s + (0.540 − 0.841i)28-s + (0.373 − 1.71i)29-s + (−0.989 − 0.142i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $-0.490 + 0.871i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2576} (1973, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2576,\ (\ :0),\ -0.490 + 0.871i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6386318363\)
\(L(\frac12)\) \(\approx\) \(0.6386318363\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.281 + 0.959i)T \)
7 \( 1 + (0.909 - 0.415i)T \)
23 \( 1 + (-0.142 + 0.989i)T \)
good3 \( 1 + (0.989 - 0.142i)T^{2} \)
5 \( 1 + (-0.281 + 0.959i)T^{2} \)
11 \( 1 + (-0.125 - 0.0683i)T + (0.540 + 0.841i)T^{2} \)
13 \( 1 + (0.755 - 0.654i)T^{2} \)
17 \( 1 + (-0.415 + 0.909i)T^{2} \)
19 \( 1 + (-0.909 + 0.415i)T^{2} \)
29 \( 1 + (-0.373 + 1.71i)T + (-0.909 - 0.415i)T^{2} \)
31 \( 1 + (0.142 - 0.989i)T^{2} \)
37 \( 1 + (-0.418 + 0.559i)T + (-0.281 - 0.959i)T^{2} \)
41 \( 1 + (-0.959 - 0.281i)T^{2} \)
43 \( 1 + (-1.59 + 0.114i)T + (0.989 - 0.142i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.334 + 0.898i)T + (-0.755 - 0.654i)T^{2} \)
59 \( 1 + (-0.755 + 0.654i)T^{2} \)
61 \( 1 + (-0.989 - 0.142i)T^{2} \)
67 \( 1 + (1.71 - 0.936i)T + (0.540 - 0.841i)T^{2} \)
71 \( 1 + (-0.368 + 1.25i)T + (-0.841 - 0.540i)T^{2} \)
73 \( 1 + (0.415 + 0.909i)T^{2} \)
79 \( 1 + (0.234 - 0.512i)T + (-0.654 - 0.755i)T^{2} \)
83 \( 1 + (-0.281 - 0.959i)T^{2} \)
89 \( 1 + (-0.142 - 0.989i)T^{2} \)
97 \( 1 + (0.959 + 0.281i)T^{2} \)
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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

−8.890649987691521275438513711982, −8.377555530435544485870234859271, −7.52368652322598431595180367603, −6.36889332796842755357870032340, −5.74711316388542027811511585408, −4.65366425808099432704549095541, −3.83407547767290887062251012318, −2.75636110949373872009296390040, −2.34872203530504890650593657149, −0.52764422555257840719761554777, 1.16962287362694638716147223229, 2.96851105867166623542024249908, 3.72615866546180190519368919506, 4.82669084821997006152285764606, 5.69072058647138228473408342143, 6.23720964551223297787356331054, 7.13118902209111350922904069386, 7.60167379053668554539954273618, 8.713566715573049521797453197599, 9.091889191687614540561938924538

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