Properties

Label 2-912-57.35-c0-0-0
Degree $2$
Conductor $912$
Sign $0.513 + 0.858i$
Analytic cond. $0.455147$
Root an. cond. $0.674646$
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  + (0.939 − 0.342i)3-s + (−0.939 − 1.62i)7-s + (0.766 − 0.642i)9-s + (−0.326 − 0.118i)13-s + (0.5 + 0.866i)19-s + (−1.43 − 1.20i)21-s + (−0.939 − 0.342i)25-s + (0.500 − 0.866i)27-s + (0.766 + 1.32i)31-s + 1.53·37-s − 0.347·39-s + (−0.266 + 1.50i)43-s + (−1.26 + 2.19i)49-s + (0.766 + 0.642i)57-s + (0.0603 + 0.342i)61-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)3-s + (−0.939 − 1.62i)7-s + (0.766 − 0.642i)9-s + (−0.326 − 0.118i)13-s + (0.5 + 0.866i)19-s + (−1.43 − 1.20i)21-s + (−0.939 − 0.342i)25-s + (0.500 − 0.866i)27-s + (0.766 + 1.32i)31-s + 1.53·37-s − 0.347·39-s + (−0.266 + 1.50i)43-s + (−1.26 + 2.19i)49-s + (0.766 + 0.642i)57-s + (0.0603 + 0.342i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.513 + 0.858i$
Analytic conductor: \(0.455147\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (833, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :0),\ 0.513 + 0.858i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.233455425\)
\(L(\frac12)\) \(\approx\) \(1.233455425\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
17 \( 1 + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - 1.53T + T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
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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

−9.814945856854973240549199312208, −9.676298836252517872237779719920, −8.278990979610009245773470563972, −7.65541011564378512372474281495, −6.91218622448607216111525748697, −6.13799428740822469199222400872, −4.50620694711157630334599757815, −3.69656193780226036249956006023, −2.85137561760291189083279524415, −1.22796604635395947468663186980, 2.25367904216824911079160065794, 2.87578115028088518878629118637, 4.00247537330916088204788532662, 5.19957207990768060614414757973, 6.06907607959908430511468906559, 7.12962072544783067088337454329, 8.112869105366693464611177768028, 8.911415952852739180944582422477, 9.546078018103092169165542708242, 9.985069270771968773449516048955

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