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Properties

Label 2-912-304.125-c1-0-12
Degree $2$
Conductor $912$
Sign $-0.942 - 0.334i$
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  + (−0.956 + 1.04i)2-s + (0.965 − 0.258i)3-s + (−0.172 − 1.99i)4-s + (1.08 + 4.03i)5-s + (−0.653 + 1.25i)6-s − 1.09i·7-s + (2.24 + 1.72i)8-s + (0.866 − 0.499i)9-s + (−5.24 − 2.73i)10-s + (−3.68 + 3.68i)11-s + (−0.681 − 1.88i)12-s + (−0.262 + 0.980i)13-s + (1.14 + 1.04i)14-s + (2.09 + 3.62i)15-s + (−3.94 + 0.685i)16-s + (−1.38 + 2.39i)17-s + ⋯
L(s)  = 1  + (−0.676 + 0.736i)2-s + (0.557 − 0.149i)3-s + (−0.0860 − 0.996i)4-s + (0.483 + 1.80i)5-s + (−0.266 + 0.511i)6-s − 0.413i·7-s + (0.792 + 0.610i)8-s + (0.288 − 0.166i)9-s + (−1.65 − 0.864i)10-s + (−1.11 + 1.11i)11-s + (−0.196 − 0.542i)12-s + (−0.0728 + 0.271i)13-s + (0.304 + 0.279i)14-s + (0.539 + 0.934i)15-s + (−0.985 + 0.171i)16-s + (−0.335 + 0.580i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.942 - 0.334i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (733, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.942 - 0.334i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.181095 + 1.05111i\)
\(L(\frac12)\) \(\approx\) \(0.181095 + 1.05111i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.956 - 1.04i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
19 \( 1 + (0.823 + 4.28i)T \)
good5 \( 1 + (-1.08 - 4.03i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + 1.09iT - 7T^{2} \)
11 \( 1 + (3.68 - 3.68i)T - 11iT^{2} \)
13 \( 1 + (0.262 - 0.980i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.38 - 2.39i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.47 - 1.43i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.22 - 4.57i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 - 8.18T + 31T^{2} \)
37 \( 1 + (-0.714 + 0.714i)T - 37iT^{2} \)
41 \( 1 + (3.45 + 1.99i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.292 - 1.09i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (5.67 + 9.82i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.34 - 1.70i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.35 - 8.79i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.03 - 7.60i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (2.80 - 10.4i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (10.3 + 5.99i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.34 - 3.08i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.08 + 7.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.56 - 3.56i)T + 83iT^{2} \)
89 \( 1 + (4.39 - 2.53i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.63 - 14.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.32259749535847695458836346742, −9.781258309340673239759396191973, −8.715264385112907773089612895076, −7.69646277703371666520825833429, −7.07332513285243176096926671894, −6.65213468310615985127875726717, −5.53007311017248016348101640159, −4.26025557239768461702494742464, −2.75675382081145830072617551273, −1.97359793295923517514624409223, 0.56293171721694068498515031092, 1.92031027835968562030528493725, 2.94858066801578912861977586802, 4.25247332303632645060949529722, 5.11877539118183468715123343085, 6.11759643122702921692015243152, 7.971359593457773443525818944234, 8.174673696813234906587937604793, 8.880340905319634435948645482157, 9.685231663148952219982233020659

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