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Properties

Label 2-2576-161.118-c0-0-0
Degree $2$
Conductor $2576$
Sign $0.105 + 0.994i$
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  + (−0.841 + 0.540i)7-s + (−0.654 − 0.755i)9-s + (0.118 − 0.822i)11-s + (0.654 − 0.755i)23-s + (−0.142 − 0.989i)25-s + (0.273 + 0.0801i)29-s + (−1.10 − 1.27i)37-s + (0.544 − 1.19i)43-s + (0.415 − 0.909i)49-s + (−0.239 + 0.153i)53-s + (0.959 + 0.281i)63-s + (−0.273 − 1.89i)67-s + (0.118 + 0.822i)71-s + (0.345 + 0.755i)77-s + (0.239 + 0.153i)79-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)7-s + (−0.654 − 0.755i)9-s + (0.118 − 0.822i)11-s + (0.654 − 0.755i)23-s + (−0.142 − 0.989i)25-s + (0.273 + 0.0801i)29-s + (−1.10 − 1.27i)37-s + (0.544 − 1.19i)43-s + (0.415 − 0.909i)49-s + (−0.239 + 0.153i)53-s + (0.959 + 0.281i)63-s + (−0.273 − 1.89i)67-s + (0.118 + 0.822i)71-s + (0.345 + 0.755i)77-s + (0.239 + 0.153i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $0.105 + 0.994i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2576} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2576,\ (\ :0),\ 0.105 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8304931095\)
\(L(\frac12)\) \(\approx\) \(0.8304931095\)
\(L(1)\) 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 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (-0.654 + 0.755i)T \)
good3 \( 1 + (0.654 + 0.755i)T^{2} \)
5 \( 1 + (0.142 + 0.989i)T^{2} \)
11 \( 1 + (-0.118 + 0.822i)T + (-0.959 - 0.281i)T^{2} \)
13 \( 1 + (-0.415 - 0.909i)T^{2} \)
17 \( 1 + (-0.841 - 0.540i)T^{2} \)
19 \( 1 + (-0.841 + 0.540i)T^{2} \)
29 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
31 \( 1 + (0.654 - 0.755i)T^{2} \)
37 \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \)
41 \( 1 + (0.142 + 0.989i)T^{2} \)
43 \( 1 + (-0.544 + 1.19i)T + (-0.654 - 0.755i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.239 - 0.153i)T + (0.415 - 0.909i)T^{2} \)
59 \( 1 + (-0.415 - 0.909i)T^{2} \)
61 \( 1 + (0.654 - 0.755i)T^{2} \)
67 \( 1 + (0.273 + 1.89i)T + (-0.959 + 0.281i)T^{2} \)
71 \( 1 + (-0.118 - 0.822i)T + (-0.959 + 0.281i)T^{2} \)
73 \( 1 + (-0.841 + 0.540i)T^{2} \)
79 \( 1 + (-0.239 - 0.153i)T + (0.415 + 0.909i)T^{2} \)
83 \( 1 + (0.142 - 0.989i)T^{2} \)
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

−8.826117457746951056037722320509, −8.450053256966831353421311833976, −7.25840160109487508731551660009, −6.41776154693309949410549155974, −5.96684368357254124124255741245, −5.11946722668930081988922766160, −3.87931387435704360673592063668, −3.17902261330908004212953501602, −2.32843679437215763337798165902, −0.54667424929354405444403634220, 1.48111447086314607804977372374, 2.73738098081506704930952407876, 3.51073774441612700795421270243, 4.56724376816364993972703472231, 5.31561322913061992761160572142, 6.21886405537105558292348104075, 7.06839340970207976038338961606, 7.59297019767204132807484959453, 8.505192596485240151705799206942, 9.338519926943278051335602443559

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