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Properties

Label 2-912-57.8-c1-0-1
Degree $2$
Conductor $912$
Sign $-0.689 - 0.724i$
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  + (−1.72 − 0.158i)3-s + (−1.22 − 0.707i)5-s + 0.449·7-s + (2.94 + 0.548i)9-s − 3.14i·11-s + (−3 + 1.73i)13-s + (1.99 + 1.41i)15-s + (−0.550 − 0.317i)17-s + (−3.17 + 2.98i)19-s + (−0.775 − 0.0714i)21-s + (6.12 − 3.53i)23-s + (−1.50 − 2.59i)25-s + (−4.99 − 1.41i)27-s + (1.22 + 2.12i)29-s + 4.24i·31-s + ⋯
L(s)  = 1  + (−0.995 − 0.0917i)3-s + (−0.547 − 0.316i)5-s + 0.169·7-s + (0.983 + 0.182i)9-s − 0.948i·11-s + (−0.832 + 0.480i)13-s + (0.516 + 0.365i)15-s + (−0.133 − 0.0770i)17-s + (−0.728 + 0.685i)19-s + (−0.169 − 0.0155i)21-s + (1.27 − 0.737i)23-s + (−0.300 − 0.519i)25-s + (−0.962 − 0.272i)27-s + (0.227 + 0.393i)29-s + 0.762i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.689 - 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.689 - 0.724i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.689 - 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0851393 + 0.198678i\)
\(L(\frac12)\) \(\approx\) \(0.0851393 + 0.198678i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 + 0.158i)T \)
19 \( 1 + (3.17 - 2.98i)T \)
good5 \( 1 + (1.22 + 0.707i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 0.449T + 7T^{2} \)
11 \( 1 + 3.14iT - 11T^{2} \)
13 \( 1 + (3 - 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.550 + 0.317i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-6.12 + 3.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.22 - 2.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 - 7.70iT - 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.44 - 7.70i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (11.5 - 6.68i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.44 + 9.43i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.72 - 9.91i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.775 - 1.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.17 - 1.25i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.39 - 7.61i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.34 - 4.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 17.0iT - 83T^{2} \)
89 \( 1 + (-3.55 - 6.14i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.84 - 1.64i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.52207902985180811481071568574, −9.731454594583957135198793623823, −8.582486166329520361450889065676, −7.929790159370875970970571086275, −6.78435024665851196463523856237, −6.22879404047453459279853661990, −4.94889830160876964978098272821, −4.51686359058294171846129145716, −3.11348767342970864004135789721, −1.39882601037000971575206727618, 0.12092908824677507958414350416, 1.96322193879449765741878637298, 3.49723176188910547344448924852, 4.64705686724673619814543687516, 5.20968726745541476963587163567, 6.42589831402705867431444599857, 7.22283923559314569643941052440, 7.77086916135327617185397335525, 9.164382285760969829326689586403, 9.889068830973434794813820434283

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