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Properties

Label 2-912-57.50-c1-0-17
Degree $2$
Conductor $912$
Sign $0.999 - 0.0179i$
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 + 1.65i)3-s + (2.18 − 1.26i)5-s − 3.37·7-s + (−2.5 − 1.65i)9-s − 3.46i·11-s + (2.87 + 1.65i)13-s + (1 + 4.25i)15-s + (2.18 − 1.26i)17-s + (4 − 1.73i)19-s + (1.68 − 5.59i)21-s + (7.93 + 4.57i)23-s + (0.686 − 1.18i)25-s + (4 − 3.31i)27-s + (−0.186 + 0.322i)29-s − 7.72i·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.957i)3-s + (0.977 − 0.564i)5-s − 1.27·7-s + (−0.833 − 0.552i)9-s − 1.04i·11-s + (0.796 + 0.459i)13-s + (0.258 + 1.09i)15-s + (0.530 − 0.306i)17-s + (0.917 − 0.397i)19-s + (0.367 − 1.22i)21-s + (1.65 + 0.954i)23-s + (0.137 − 0.237i)25-s + (0.769 − 0.638i)27-s + (−0.0345 + 0.0598i)29-s − 1.38i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.999 - 0.0179i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.999 - 0.0179i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50772 + 0.0134966i\)
\(L(\frac12)\) \(\approx\) \(1.50772 + 0.0134966i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 1.65i)T \)
19 \( 1 + (-4 + 1.73i)T \)
good5 \( 1 + (-2.18 + 1.26i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.37T + 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + (-2.87 - 1.65i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.18 + 1.26i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-7.93 - 4.57i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.186 - 0.322i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.72iT - 31T^{2} \)
37 \( 1 + 11.1iT - 37T^{2} \)
41 \( 1 + (-3.18 - 5.51i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.87 - 10.1i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.813 + 0.469i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.81 - 4.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.813 + 1.40i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.24 - 0.718i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.18 + 10.7i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.87 + 4.97i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-12.9 + 7.49i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 + (0.813 - 1.40i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.44 + 1.40i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.797732075067669589229870056131, −9.270871991507529608534615295414, −9.040210573114173729953037844010, −7.55248517864707559301896974335, −6.15755990880469641101682424949, −5.88011070414786089192942119330, −4.93302925441963677960241402278, −3.63345108687961436578393319518, −2.92498393573560519748166763561, −0.905688510322046202803204416781, 1.20312932012362480941499771938, 2.54393436579095325077830446798, 3.36061022765361375452663154638, 5.14559467704018874175874733711, 5.96815269680143070629186166598, 6.71938448756814750981145849552, 7.17285942522205425486296098798, 8.399752719197876835621716635077, 9.344091836795778145203780246234, 10.22956598199647540226676154327

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