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Properties

Label 2-912-12.11-c1-0-18
Degree $2$
Conductor $912$
Sign $0.848 + 0.529i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.916 − 1.46i)3-s + 0.860i·5-s + 2.31i·7-s + (−1.31 − 2.69i)9-s + 0.512·11-s + 2.93·13-s + (1.26 + 0.789i)15-s − 5.22i·17-s + i·19-s + (3.40 + 2.12i)21-s + 9.29·23-s + 4.25·25-s + (−5.16 − 0.530i)27-s − 6.87i·29-s + 4.93i·31-s + ⋯
L(s)  = 1  + (0.529 − 0.848i)3-s + 0.385i·5-s + 0.876i·7-s + (−0.439 − 0.898i)9-s + 0.154·11-s + 0.815·13-s + (0.326 + 0.203i)15-s − 1.26i·17-s + 0.229i·19-s + (0.743 + 0.463i)21-s + 1.93·23-s + 0.851·25-s + (−0.994 − 0.102i)27-s − 1.27i·29-s + 0.887i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.848 + 0.529i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.848 + 0.529i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90908 - 0.546571i\)
\(L(\frac12)\) \(\approx\) \(1.90908 - 0.546571i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.916 + 1.46i)T \)
19 \( 1 - iT \)
good5 \( 1 - 0.860iT - 5T^{2} \)
7 \( 1 - 2.31iT - 7T^{2} \)
11 \( 1 - 0.512T + 11T^{2} \)
13 \( 1 - 2.93T + 13T^{2} \)
17 \( 1 + 5.22iT - 17T^{2} \)
23 \( 1 - 9.29T + 23T^{2} \)
29 \( 1 + 6.87iT - 29T^{2} \)
31 \( 1 - 4.93iT - 31T^{2} \)
37 \( 1 - 9.45T + 37T^{2} \)
41 \( 1 - 6.43iT - 41T^{2} \)
43 \( 1 - 1.68iT - 43T^{2} \)
47 \( 1 + 6.59T + 47T^{2} \)
53 \( 1 + 3.31iT - 53T^{2} \)
59 \( 1 + 1.13T + 59T^{2} \)
61 \( 1 + 2.01T + 61T^{2} \)
67 \( 1 + 7.15iT - 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 + 4.49T + 73T^{2} \)
79 \( 1 - 5.36iT - 79T^{2} \)
83 \( 1 + 8.96T + 83T^{2} \)
89 \( 1 + 1.04iT - 89T^{2} \)
97 \( 1 + 8.21T + 97T^{2} \)
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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

−9.768024912842956055558839869307, −9.037761643058499526982860746569, −8.412713944088977714059860436136, −7.43905265318565459584817830208, −6.65917053119675261770710907741, −5.91140801256690017211939003304, −4.74948028531354498309519527550, −3.19409364609558164625361083550, −2.62077771220451871473041703154, −1.14507806997805077080897539392, 1.27522048150559850301708049510, 2.96083240848835669225183568813, 3.91093279573322624152565793336, 4.63950251879815859787189301002, 5.66637540900557307107562405406, 6.83737502686928913001473390983, 7.78552482180234704106846932723, 8.761081955365791510302252022829, 9.118517800451916617889100905818, 10.30888227942593251177486130919

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