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Properties

Label 2-52e2-676.311-c0-0-0
Degree $2$
Conductor $2704$
Sign $0.975 - 0.221i$
Analytic cond. $1.34947$
Root an. cond. $1.16166$
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  + (−1.85 + 0.225i)5-s + (−0.354 − 0.935i)9-s + (−0.885 + 0.464i)13-s + (1.32 + 0.695i)17-s + (2.42 − 0.597i)25-s + (0.402 + 1.06i)29-s + (0.475 − 1.92i)37-s + (1.53 + 1.06i)41-s + (0.869 + 1.65i)45-s + (−0.568 − 0.822i)49-s + (1.56 + 0.822i)53-s + (1.00 − 0.527i)61-s + (1.53 − 1.06i)65-s + (−0.748 + 0.663i)81-s + (−2.61 − 0.992i)85-s + ⋯
L(s)  = 1  + (−1.85 + 0.225i)5-s + (−0.354 − 0.935i)9-s + (−0.885 + 0.464i)13-s + (1.32 + 0.695i)17-s + (2.42 − 0.597i)25-s + (0.402 + 1.06i)29-s + (0.475 − 1.92i)37-s + (1.53 + 1.06i)41-s + (0.869 + 1.65i)45-s + (−0.568 − 0.822i)49-s + (1.56 + 0.822i)53-s + (1.00 − 0.527i)61-s + (1.53 − 1.06i)65-s + (−0.748 + 0.663i)81-s + (−2.61 − 0.992i)85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2704\)    =    \(2^{4} \cdot 13^{2}\)
Sign: $0.975 - 0.221i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2704} (1663, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2704,\ (\ :0),\ 0.975 - 0.221i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7633596798\)
\(L(\frac12)\) \(\approx\) \(0.7633596798\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (0.885 - 0.464i)T \)
good3 \( 1 + (0.354 + 0.935i)T^{2} \)
5 \( 1 + (1.85 - 0.225i)T + (0.970 - 0.239i)T^{2} \)
7 \( 1 + (0.568 + 0.822i)T^{2} \)
11 \( 1 + (-0.748 - 0.663i)T^{2} \)
17 \( 1 + (-1.32 - 0.695i)T + (0.568 + 0.822i)T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.402 - 1.06i)T + (-0.748 + 0.663i)T^{2} \)
31 \( 1 + (0.885 + 0.464i)T^{2} \)
37 \( 1 + (-0.475 + 1.92i)T + (-0.885 - 0.464i)T^{2} \)
41 \( 1 + (-1.53 - 1.06i)T + (0.354 + 0.935i)T^{2} \)
43 \( 1 + (-0.885 + 0.464i)T^{2} \)
47 \( 1 + (0.120 + 0.992i)T^{2} \)
53 \( 1 + (-1.56 - 0.822i)T + (0.568 + 0.822i)T^{2} \)
59 \( 1 + (-0.970 + 0.239i)T^{2} \)
61 \( 1 + (-1.00 + 0.527i)T + (0.568 - 0.822i)T^{2} \)
67 \( 1 + (0.120 + 0.992i)T^{2} \)
71 \( 1 + (-0.354 - 0.935i)T^{2} \)
73 \( 1 + (0.748 + 0.663i)T^{2} \)
79 \( 1 + (-0.120 - 0.992i)T^{2} \)
83 \( 1 + (-0.354 + 0.935i)T^{2} \)
89 \( 1 - 1.98iT - T^{2} \)
97 \( 1 + (0.475 + 0.0576i)T + (0.970 + 0.239i)T^{2} \)
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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

−8.925749772185836912483486887199, −8.202236960764748726348490272412, −7.51131195023939618754915428768, −7.01097608940711662274603578652, −6.05617532760586803310964728243, −5.04030652343544113474279155714, −4.03942025252913049006375917762, −3.59736249514218508386348257497, −2.66307592398096078849007622444, −0.854458141223483994231146238741, 0.74534884107564532729417428327, 2.57373146998593214445083696194, 3.31542226525970034363571598303, 4.34414915792876546404556693484, 4.92223960752483315508015296716, 5.75278126274891054472466653620, 7.07991540706924933900434683105, 7.64257787454720823693227587407, 8.043501218937152497832775081015, 8.725382498459715058494693823760

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