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Properties

Label 2-912-57.29-c1-0-23
Degree $2$
Conductor $912$
Sign $0.962 - 0.271i$
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  + (1.69 + 0.355i)3-s + (−0.882 + 2.42i)5-s + (1.58 − 2.74i)7-s + (2.74 + 1.20i)9-s + (2.16 − 1.25i)11-s + (2.71 − 3.24i)13-s + (−2.35 + 3.79i)15-s + (1.32 − 0.233i)17-s + (−3.14 − 3.01i)19-s + (3.65 − 4.08i)21-s + (1.30 + 3.58i)23-s + (−1.27 − 1.06i)25-s + (4.23 + 3.01i)27-s + (−1.32 + 7.49i)29-s + (−6.89 − 3.97i)31-s + ⋯
L(s)  = 1  + (0.978 + 0.205i)3-s + (−0.394 + 1.08i)5-s + (0.598 − 1.03i)7-s + (0.915 + 0.401i)9-s + (0.653 − 0.377i)11-s + (0.754 − 0.898i)13-s + (−0.608 + 0.980i)15-s + (0.320 − 0.0565i)17-s + (−0.722 − 0.691i)19-s + (0.798 − 0.892i)21-s + (0.272 + 0.747i)23-s + (−0.254 − 0.213i)25-s + (0.814 + 0.580i)27-s + (−0.245 + 1.39i)29-s + (−1.23 − 0.714i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.962 - 0.271i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.962 - 0.271i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.35671 + 0.325795i\)
\(L(\frac12)\) \(\approx\) \(2.35671 + 0.325795i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 - 0.355i)T \)
19 \( 1 + (3.14 + 3.01i)T \)
good5 \( 1 + (0.882 - 2.42i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (-1.58 + 2.74i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.16 + 1.25i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.71 + 3.24i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-1.32 + 0.233i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (-1.30 - 3.58i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (1.32 - 7.49i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (6.89 + 3.97i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.10iT - 37T^{2} \)
41 \( 1 + (-4.95 + 4.16i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-11.7 - 4.27i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (6.16 + 1.08i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (3.46 - 1.26i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-1.54 - 8.75i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (0.133 - 0.0485i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-4.48 - 0.791i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (8.59 + 3.12i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (1.67 - 1.40i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (6.41 + 7.64i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (12.3 + 7.11i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (12.7 + 10.6i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-0.538 + 0.0949i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.28063237632318371233755193698, −9.197170736839759792559950732287, −8.436952918588328731686256243705, −7.38139933958057251531268580863, −7.22268198498548523724251340548, −5.88138361383336206291687887841, −4.44859927785974918653928781468, −3.62726719366061123702955456942, −2.94859359365276064872127434552, −1.35775881659542480504113826135, 1.39753797131589415689109081052, 2.30560047880278329485065551663, 3.89962058803181403271247356901, 4.44903022040941333049838043066, 5.68018656122766986580571559396, 6.71476616160784036742787128325, 7.86494085696518762961734353261, 8.507523224459638296968640261450, 9.001988050079073885190724947230, 9.637151465763506450713105190878

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