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Properties

Label 2-912-57.5-c0-0-0
Degree $2$
Conductor $912$
Sign $-0.0389 - 0.999i$
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  + (−0.173 + 0.984i)3-s + (0.173 + 0.300i)7-s + (−0.939 − 0.342i)9-s + (0.266 + 1.50i)13-s + (0.5 + 0.866i)19-s + (−0.326 + 0.118i)21-s + (0.173 + 0.984i)25-s + (0.5 − 0.866i)27-s + (−0.939 − 1.62i)31-s − 1.87·37-s − 1.53·39-s + (1.43 + 1.20i)43-s + (0.439 − 0.761i)49-s + (−0.939 + 0.342i)57-s + (1.17 − 0.984i)61-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)3-s + (0.173 + 0.300i)7-s + (−0.939 − 0.342i)9-s + (0.266 + 1.50i)13-s + (0.5 + 0.866i)19-s + (−0.326 + 0.118i)21-s + (0.173 + 0.984i)25-s + (0.5 − 0.866i)27-s + (−0.939 − 1.62i)31-s − 1.87·37-s − 1.53·39-s + (1.43 + 1.20i)43-s + (0.439 − 0.761i)49-s + (−0.939 + 0.342i)57-s + (1.17 − 0.984i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0389 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0389 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.0389 - 0.999i$
Analytic conductor: \(0.455147\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :0),\ -0.0389 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9058642966\)
\(L(\frac12)\) \(\approx\) \(0.9058642966\)
\(L(1)\) 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 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.173 - 0.984i)T^{2} \)
7 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.766 + 0.642i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + 1.87T + T^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (-1.43 - 1.20i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 + (-0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.939 - 0.342i)T^{2} \)
97 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
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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

−10.51604583362750403729811193967, −9.496293980369288987932063415757, −9.146813690539272225509266118348, −8.163984743616267524375787858247, −7.07090946578635783078042349800, −6.00398955127023013067822480762, −5.25422472975716030756694238752, −4.21334276750321749810035647422, −3.45249173387382402618476299658, −1.95280307646931823337428521543, 0.969150725369289885523021710750, 2.46609609339575195967818244721, 3.52335292919550285782092668387, 5.06941569722518177602294777764, 5.71193516494705598433745286602, 6.85928637186794851178538533263, 7.41870320647943900133673139382, 8.353082198407246230210061242536, 8.981616874052159909630697282089, 10.43093724485241708136471924493

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