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Properties

Label 2-912-19.5-c1-0-1
Degree $2$
Conductor $912$
Sign $0.138 - 0.990i$
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  + (0.173 − 0.984i)3-s + (−0.328 − 0.276i)5-s + (1.95 + 3.38i)7-s + (−0.939 − 0.342i)9-s + (−2.16 + 3.75i)11-s + (−0.0235 − 0.133i)13-s + (−0.328 + 0.276i)15-s + (−5.59 + 2.03i)17-s + (1.47 + 4.10i)19-s + (3.67 − 1.33i)21-s + (−6.07 + 5.09i)23-s + (−0.836 − 4.74i)25-s + (−0.5 + 0.866i)27-s + (7.20 + 2.62i)29-s + (−2.60 − 4.51i)31-s + ⋯
L(s)  = 1  + (0.100 − 0.568i)3-s + (−0.147 − 0.123i)5-s + (0.738 + 1.27i)7-s + (−0.313 − 0.114i)9-s + (−0.653 + 1.13i)11-s + (−0.00653 − 0.0370i)13-s + (−0.0849 + 0.0712i)15-s + (−1.35 + 0.494i)17-s + (0.338 + 0.940i)19-s + (0.801 − 0.291i)21-s + (−1.26 + 1.06i)23-s + (−0.167 − 0.948i)25-s + (−0.0962 + 0.166i)27-s + (1.33 + 0.487i)29-s + (−0.467 − 0.810i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.138 - 0.990i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.138 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.911035 + 0.792422i\)
\(L(\frac12)\) \(\approx\) \(0.911035 + 0.792422i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (-1.47 - 4.10i)T \)
good5 \( 1 + (0.328 + 0.276i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-1.95 - 3.38i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.16 - 3.75i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.0235 + 0.133i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (5.59 - 2.03i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (6.07 - 5.09i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-7.20 - 2.62i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (2.60 + 4.51i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.10T + 37T^{2} \)
41 \( 1 + (0.555 - 3.15i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-7.94 - 6.66i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (3.56 + 1.29i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-5.49 + 4.60i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (0.0603 - 0.0219i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-4.56 + 3.82i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-15.0 - 5.46i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-5.18 - 4.34i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (1.71 - 9.71i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-1.77 + 10.0i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (3.33 + 5.76i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.18 - 12.4i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-7.87 + 2.86i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.20543462774204159438135594274, −9.399339659303849150508154118441, −8.256297594601005868335781001542, −8.086285745979136742000473119585, −6.90468464679732527976716881236, −5.92659026961139413518187237187, −5.11543796056980221535989815861, −4.08158555933456240540861469253, −2.45429922454994703583516237104, −1.82177363572537154442890040782, 0.54486036543808720179631631957, 2.42531697387596523937434597832, 3.65190585219932156782615988900, 4.50316801569376822116437906369, 5.29753041204651993238270714783, 6.57607513709002218677346195725, 7.40442985639790386210350302594, 8.322776055562766258449250289501, 8.943633431782376675424846533235, 10.11187960200727877967881473484

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