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Properties

Label 2-912-19.11-c1-0-14
Degree $2$
Conductor $912$
Sign $0.485 + 0.874i$
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  + (0.5 − 0.866i)3-s + (1.33 − 2.31i)5-s + 3.67·7-s + (−0.499 − 0.866i)9-s + 3.81·11-s + (−0.0719 − 0.124i)13-s + (−1.33 − 2.31i)15-s + (−4.24 − 0.990i)19-s + (1.83 − 3.18i)21-s + (3.76 + 6.52i)23-s + (−1.07 − 1.85i)25-s − 0.999·27-s + (2.67 + 4.62i)29-s − 8.81·31-s + (1.90 − 3.30i)33-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.597 − 1.03i)5-s + 1.38·7-s + (−0.166 − 0.288i)9-s + 1.15·11-s + (−0.0199 − 0.0345i)13-s + (−0.345 − 0.597i)15-s + (−0.973 − 0.227i)19-s + (0.400 − 0.694i)21-s + (0.784 + 1.35i)23-s + (−0.214 − 0.371i)25-s − 0.192·27-s + (0.496 + 0.859i)29-s − 1.58·31-s + (0.332 − 0.575i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.485 + 0.874i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.485 + 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98943 - 1.17050i\)
\(L(\frac12)\) \(\approx\) \(1.98943 - 1.17050i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (4.24 + 0.990i)T \)
good5 \( 1 + (-1.33 + 2.31i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.67T + 7T^{2} \)
11 \( 1 - 3.81T + 11T^{2} \)
13 \( 1 + (0.0719 + 0.124i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.76 - 6.52i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.67 - 4.62i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.81T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (2.67 - 4.62i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.40 + 2.43i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.00 + 6.94i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.90 + 3.30i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.74 + 9.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.69 - 4.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.81 - 11.8i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.172 + 0.299i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.26 + 5.65i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.28T + 83T^{2} \)
89 \( 1 + (-4.33 - 7.51i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.95 + 5.12i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.687602445260114606273962647418, −8.898184799462196020115279225251, −8.503103355968150263114250909784, −7.48045977610309180578869957956, −6.58622649637564513378121975557, −5.41107166601271901234553245980, −4.80860422408671284565946724250, −3.61896890045984704043603794984, −1.87335476968152249903907577889, −1.31163375389445785080215730663, 1.70398043451543250361502145971, 2.70808361790408769566856713064, 4.01358151081215318121851385750, 4.78496160871144732162276814711, 5.99866644849616802944813079689, 6.75619879491873938563068586270, 7.75936140975468510766284008072, 8.695595420410064006317286560542, 9.321136161266522709633899826947, 10.52816180307033288232081455851

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