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Properties

Label 2-3104-776.579-c0-0-0
Degree $2$
Conductor $3104$
Sign $0.645 + 0.763i$
Analytic cond. $1.54909$
Root an. cond. $1.24462$
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  + (−1.20 − 0.158i)3-s + (0.465 + 0.124i)9-s + (−1.17 + 1.53i)11-s + (−0.357 − 1.05i)17-s + (0.382 − 1.92i)19-s + (0.130 + 0.991i)25-s + (0.582 + 0.241i)27-s + (1.66 − 1.66i)33-s + (−0.0255 − 0.389i)41-s + (1.91 − 0.513i)43-s + (0.793 + 0.608i)49-s + (0.263 + 1.32i)51-s + (−0.767 + 2.26i)57-s + (−0.735 − 1.49i)59-s + (0.128 + 0.0255i)67-s + ⋯
L(s)  = 1  + (−1.20 − 0.158i)3-s + (0.465 + 0.124i)9-s + (−1.17 + 1.53i)11-s + (−0.357 − 1.05i)17-s + (0.382 − 1.92i)19-s + (0.130 + 0.991i)25-s + (0.582 + 0.241i)27-s + (1.66 − 1.66i)33-s + (−0.0255 − 0.389i)41-s + (1.91 − 0.513i)43-s + (0.793 + 0.608i)49-s + (0.263 + 1.32i)51-s + (−0.767 + 2.26i)57-s + (−0.735 − 1.49i)59-s + (0.128 + 0.0255i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3104\)    =    \(2^{5} \cdot 97\)
Sign: $0.645 + 0.763i$
Analytic conductor: \(1.54909\)
Root analytic conductor: \(1.24462\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3104} (1743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3104,\ (\ :0),\ 0.645 + 0.763i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6127446374\)
\(L(\frac12)\) \(\approx\) \(0.6127446374\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
97 \( 1 + (-0.991 - 0.130i)T \)
good3 \( 1 + (1.20 + 0.158i)T + (0.965 + 0.258i)T^{2} \)
5 \( 1 + (-0.130 - 0.991i)T^{2} \)
7 \( 1 + (-0.793 - 0.608i)T^{2} \)
11 \( 1 + (1.17 - 1.53i)T + (-0.258 - 0.965i)T^{2} \)
13 \( 1 + (0.130 + 0.991i)T^{2} \)
17 \( 1 + (0.357 + 1.05i)T + (-0.793 + 0.608i)T^{2} \)
19 \( 1 + (-0.382 + 1.92i)T + (-0.923 - 0.382i)T^{2} \)
23 \( 1 + (0.608 + 0.793i)T^{2} \)
29 \( 1 + (-0.991 + 0.130i)T^{2} \)
31 \( 1 + (0.965 + 0.258i)T^{2} \)
37 \( 1 + (-0.608 + 0.793i)T^{2} \)
41 \( 1 + (0.0255 + 0.389i)T + (-0.991 + 0.130i)T^{2} \)
43 \( 1 + (-1.91 + 0.513i)T + (0.866 - 0.5i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (0.258 - 0.965i)T^{2} \)
59 \( 1 + (0.735 + 1.49i)T + (-0.608 + 0.793i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.128 - 0.0255i)T + (0.923 + 0.382i)T^{2} \)
71 \( 1 + (0.991 + 0.130i)T^{2} \)
73 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.707 + 0.707i)T^{2} \)
83 \( 1 + (-1.69 + 0.576i)T + (0.793 - 0.608i)T^{2} \)
89 \( 1 + (0.0999 + 0.241i)T + (-0.707 + 0.707i)T^{2} \)
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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

−9.064113439255463073731553361634, −7.62388050746867452009392159815, −7.26407981494712666190940949133, −6.59993443684376841896732080092, −5.57030714628054103495158076847, −4.95672417875904069999531961061, −4.54992673939177369052001983576, −2.99535726560672902421065261412, −2.14729990935756365874884120293, −0.56227069530545432995740059242, 0.953658272059831046423545549700, 2.46169330247526200483843219064, 3.53684198985613688260036575095, 4.41761473844007297160262335679, 5.44838734703180866555951216326, 5.90953583971881628325245974461, 6.30621032830821716465085268737, 7.59266579118080406475494814212, 8.207734370745482847742486541180, 8.820541639529236783360850651085

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