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Properties

Label 2-2883-93.56-c0-0-0
Degree $2$
Conductor $2883$
Sign $-0.978 + 0.205i$
Analytic cond. $1.43880$
Root an. cond. $1.19950$
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  + i·2-s + (−0.258 + 0.965i)3-s + (0.866 + 0.5i)5-s + (−0.965 − 0.258i)6-s + (0.5 + 0.866i)7-s + i·8-s + (−0.866 − 0.499i)9-s + (−0.5 + 0.866i)10-s + (−0.866 + 0.5i)14-s + (−0.707 + 0.707i)15-s − 16-s + (−1.22 + 0.707i)17-s + (0.499 − 0.866i)18-s + (−0.5 − 0.866i)19-s + (−0.965 + 0.258i)21-s + ⋯
L(s)  = 1  + i·2-s + (−0.258 + 0.965i)3-s + (0.866 + 0.5i)5-s + (−0.965 − 0.258i)6-s + (0.5 + 0.866i)7-s + i·8-s + (−0.866 − 0.499i)9-s + (−0.5 + 0.866i)10-s + (−0.866 + 0.5i)14-s + (−0.707 + 0.707i)15-s − 16-s + (−1.22 + 0.707i)17-s + (0.499 − 0.866i)18-s + (−0.5 − 0.866i)19-s + (−0.965 + 0.258i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2883\)    =    \(3 \cdot 31^{2}\)
Sign: $-0.978 + 0.205i$
Analytic conductor: \(1.43880\)
Root analytic conductor: \(1.19950\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2883} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2883,\ (\ :0),\ -0.978 + 0.205i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.400268327\)
\(L(\frac12)\) \(\approx\) \(1.400268327\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.258 - 0.965i)T \)
31 \( 1 \)
good2 \( 1 - iT - T^{2} \)
5 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−9.226232116460775402095878188041, −8.581814102962874847517606022213, −7.959133448924433574966136967416, −6.76905504669273908142330431204, −6.17664168577392467627317122779, −5.77943994071841724689114172487, −4.89809493016854334306431979543, −4.19694240114209942550036228802, −2.63149506329882584418219708153, −2.21283053463287959866351838670, 0.880612656003352548568841292378, 1.79319824322268227631990662280, 2.34473755578949472059345903554, 3.57219628387926544467053978881, 4.58835669008945153929905460699, 5.47505245590347715810511696144, 6.36754816570023133241645531525, 7.03888987813915937850200316505, 7.66406103399677022412691830905, 8.731857967598501643418906162838

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