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Properties

Label 2-2576-161.27-c0-0-0
Degree $2$
Conductor $2576$
Sign $0.952 + 0.305i$
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  + (0.142 + 0.989i)7-s + (0.841 − 0.540i)9-s + (0.797 − 1.74i)11-s + (−0.841 − 0.540i)23-s + (0.415 + 0.909i)25-s + (−0.544 − 0.627i)29-s + (−0.239 + 0.153i)37-s + (1.61 − 0.474i)43-s + (−0.959 + 0.281i)49-s + (−0.118 − 0.822i)53-s + (0.654 + 0.755i)63-s + (0.544 + 1.19i)67-s + (0.797 + 1.74i)71-s + (1.84 + 0.540i)77-s + (0.118 − 0.822i)79-s + ⋯
L(s)  = 1  + (0.142 + 0.989i)7-s + (0.841 − 0.540i)9-s + (0.797 − 1.74i)11-s + (−0.841 − 0.540i)23-s + (0.415 + 0.909i)25-s + (−0.544 − 0.627i)29-s + (−0.239 + 0.153i)37-s + (1.61 − 0.474i)43-s + (−0.959 + 0.281i)49-s + (−0.118 − 0.822i)53-s + (0.654 + 0.755i)63-s + (0.544 + 1.19i)67-s + (0.797 + 1.74i)71-s + (1.84 + 0.540i)77-s + (0.118 − 0.822i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $0.952 + 0.305i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2576} (993, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2576,\ (\ :0),\ 0.952 + 0.305i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.361678350\)
\(L(\frac12)\) \(\approx\) \(1.361678350\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.142 - 0.989i)T \)
23 \( 1 + (0.841 + 0.540i)T \)
good3 \( 1 + (-0.841 + 0.540i)T^{2} \)
5 \( 1 + (-0.415 - 0.909i)T^{2} \)
11 \( 1 + (-0.797 + 1.74i)T + (-0.654 - 0.755i)T^{2} \)
13 \( 1 + (0.959 + 0.281i)T^{2} \)
17 \( 1 + (0.142 - 0.989i)T^{2} \)
19 \( 1 + (0.142 + 0.989i)T^{2} \)
29 \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \)
31 \( 1 + (-0.841 - 0.540i)T^{2} \)
37 \( 1 + (0.239 - 0.153i)T + (0.415 - 0.909i)T^{2} \)
41 \( 1 + (-0.415 - 0.909i)T^{2} \)
43 \( 1 + (-1.61 + 0.474i)T + (0.841 - 0.540i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.118 + 0.822i)T + (-0.959 + 0.281i)T^{2} \)
59 \( 1 + (0.959 + 0.281i)T^{2} \)
61 \( 1 + (-0.841 - 0.540i)T^{2} \)
67 \( 1 + (-0.544 - 1.19i)T + (-0.654 + 0.755i)T^{2} \)
71 \( 1 + (-0.797 - 1.74i)T + (-0.654 + 0.755i)T^{2} \)
73 \( 1 + (0.142 + 0.989i)T^{2} \)
79 \( 1 + (-0.118 + 0.822i)T + (-0.959 - 0.281i)T^{2} \)
83 \( 1 + (-0.415 + 0.909i)T^{2} \)
89 \( 1 + (-0.841 + 0.540i)T^{2} \)
97 \( 1 + (-0.415 - 0.909i)T^{2} \)
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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

−8.958917269767867778607236988233, −8.474485897955268977026348962912, −7.55619479000791128867393246103, −6.56264151618139293291329358561, −6.00033729962757622646848219880, −5.28818198690844608631800134249, −4.08509921989312878732239927032, −3.44680716595797936507381812347, −2.32586170648228929779820643939, −1.07483423817986730879784583608, 1.37722536606557823010947093849, 2.18366396622966828759136081025, 3.71194552963193452447945131495, 4.38128160572623231588839142935, 4.89928644476787090170549652998, 6.18549484106464089989886036340, 7.06245885811752961188360002782, 7.41317219499141725964556860233, 8.160031464091750697404640746780, 9.406254164564024966562052516946

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