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Properties

Label 2-2576-161.62-c0-0-0
Degree $2$
Conductor $2576$
Sign $0.988 + 0.153i$
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  + (0.959 − 0.281i)7-s + (0.415 − 0.909i)9-s + (1.10 + 1.27i)11-s + (−0.415 − 0.909i)23-s + (−0.654 + 0.755i)25-s + (0.186 − 1.29i)29-s + (−0.797 + 1.74i)37-s + (−0.698 + 0.449i)43-s + (0.841 − 0.540i)49-s + (1.25 − 0.368i)53-s + (0.142 − 0.989i)63-s + (−0.186 + 0.215i)67-s + (1.10 − 1.27i)71-s + (1.41 + 0.909i)77-s + (−1.25 − 0.368i)79-s + ⋯
L(s)  = 1  + (0.959 − 0.281i)7-s + (0.415 − 0.909i)9-s + (1.10 + 1.27i)11-s + (−0.415 − 0.909i)23-s + (−0.654 + 0.755i)25-s + (0.186 − 1.29i)29-s + (−0.797 + 1.74i)37-s + (−0.698 + 0.449i)43-s + (0.841 − 0.540i)49-s + (1.25 − 0.368i)53-s + (0.142 − 0.989i)63-s + (−0.186 + 0.215i)67-s + (1.10 − 1.27i)71-s + (1.41 + 0.909i)77-s + (−1.25 − 0.368i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $0.988 + 0.153i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2576} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2576,\ (\ :0),\ 0.988 + 0.153i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.470234421\)
\(L(\frac12)\) \(\approx\) \(1.470234421\)
\(L(1)\) 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 \( 1 + (-0.959 + 0.281i)T \)
23 \( 1 + (0.415 + 0.909i)T \)
good3 \( 1 + (-0.415 + 0.909i)T^{2} \)
5 \( 1 + (0.654 - 0.755i)T^{2} \)
11 \( 1 + (-1.10 - 1.27i)T + (-0.142 + 0.989i)T^{2} \)
13 \( 1 + (-0.841 - 0.540i)T^{2} \)
17 \( 1 + (0.959 + 0.281i)T^{2} \)
19 \( 1 + (0.959 - 0.281i)T^{2} \)
29 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
31 \( 1 + (-0.415 - 0.909i)T^{2} \)
37 \( 1 + (0.797 - 1.74i)T + (-0.654 - 0.755i)T^{2} \)
41 \( 1 + (0.654 - 0.755i)T^{2} \)
43 \( 1 + (0.698 - 0.449i)T + (0.415 - 0.909i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-1.25 + 0.368i)T + (0.841 - 0.540i)T^{2} \)
59 \( 1 + (-0.841 - 0.540i)T^{2} \)
61 \( 1 + (-0.415 - 0.909i)T^{2} \)
67 \( 1 + (0.186 - 0.215i)T + (-0.142 - 0.989i)T^{2} \)
71 \( 1 + (-1.10 + 1.27i)T + (-0.142 - 0.989i)T^{2} \)
73 \( 1 + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (1.25 + 0.368i)T + (0.841 + 0.540i)T^{2} \)
83 \( 1 + (0.654 + 0.755i)T^{2} \)
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

−9.064349502451675215926716015590, −8.304756626464170426422216511430, −7.46940540246024075094776318900, −6.78646426883187689117264250392, −6.14389621785781258481709836617, −4.91316453068783867095527068137, −4.30903023426450463959870226930, −3.59563961822684095253747044764, −2.11983740243523639836700254972, −1.26037426344415934203720008828, 1.33800854083955477186566192082, 2.23076198920744918440890718417, 3.54628214658172933086685576062, 4.27738294250448295159509585676, 5.32315003556977256360719892873, 5.80182624428223781444907543210, 6.90895405834951059105558295614, 7.60461660722681361898180895459, 8.486658586079604906137431881199, 8.815708907455776789842804124907

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