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Properties

Label 2-2576-2576.1539-c0-0-1
Degree $2$
Conductor $2576$
Sign $0.999 - 0.0415i$
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 + (0.340 − 1.56i)11-s + (−0.654 − 0.755i)14-s + (0.841 + 0.540i)16-s + (0.142 + 0.989i)18-s + (0.559 − 1.50i)22-s + (0.959 + 0.281i)23-s + (−0.540 + 0.841i)25-s + (−0.540 − 0.841i)28-s + (1.83 − 0.682i)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 + (0.340 − 1.56i)11-s + (−0.654 − 0.755i)14-s + (0.841 + 0.540i)16-s + (0.142 + 0.989i)18-s + (0.559 − 1.50i)22-s + (0.959 + 0.281i)23-s + (−0.540 + 0.841i)25-s + (−0.540 − 0.841i)28-s + (1.83 − 0.682i)29-s + (0.755 + 0.654i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $0.999 - 0.0415i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2576} (1539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2576,\ (\ :0),\ 0.999 - 0.0415i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.274616061\)
\(L(\frac12)\) \(\approx\) \(2.274616061\)
\(L(1)\) 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 \( 1 + (-0.340 + 1.56i)T + (-0.909 - 0.415i)T^{2} \)
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 \( 1 + (-1.83 + 0.682i)T + (0.755 - 0.654i)T^{2} \)
31 \( 1 + (-0.959 - 0.281i)T^{2} \)
37 \( 1 + (0.125 + 0.0683i)T + (0.540 + 0.841i)T^{2} \)
41 \( 1 + (0.841 + 0.540i)T^{2} \)
43 \( 1 + (0.767 + 0.574i)T + (0.281 + 0.959i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (1.94 - 0.139i)T + (0.989 - 0.142i)T^{2} \)
59 \( 1 + (0.989 + 0.142i)T^{2} \)
61 \( 1 + (0.281 - 0.959i)T^{2} \)
67 \( 1 + (-0.398 - 1.83i)T + (-0.909 + 0.415i)T^{2} \)
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

−8.960089687625013664577853454033, −8.107259505112312191203531049214, −7.41147874397419432089486491461, −6.62197313040766535893752974517, −5.99265236895886390846800092298, −5.14572612063863946924152353709, −4.30677823276020477199391966750, −3.40560555362041319143636483346, −2.80049598356866254381063356572, −1.34283032320269717580593352141, 1.47820122125363717541183638531, 2.63322789547736826512231952286, 3.36535218956912938112502292999, 4.40030647015829417310278078680, 4.94072685202592570434457637279, 6.10430915693213117474646692352, 6.64210526427229000575653162871, 7.11091282820109645946093616502, 8.271353560597445697235123077042, 9.295799129947053057721660716779

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