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Properties

Label 2-912-76.51-c1-0-2
Degree $2$
Conductor $912$
Sign $0.744 - 0.667i$
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  + (−0.939 − 0.342i)3-s + (−0.0537 − 0.304i)5-s + (−4.22 − 2.43i)7-s + (0.766 + 0.642i)9-s + (−1.67 + 0.966i)11-s + (1.49 + 4.11i)13-s + (−0.0537 + 0.304i)15-s + (−3.57 + 2.99i)17-s + (3.42 + 2.69i)19-s + (3.13 + 3.73i)21-s + (7.72 + 1.36i)23-s + (4.60 − 1.67i)25-s + (−0.500 − 0.866i)27-s + (5.51 − 6.57i)29-s + (1.67 − 2.89i)31-s + ⋯
L(s)  = 1  + (−0.542 − 0.197i)3-s + (−0.0240 − 0.136i)5-s + (−1.59 − 0.922i)7-s + (0.255 + 0.214i)9-s + (−0.504 + 0.291i)11-s + (0.415 + 1.14i)13-s + (−0.0138 + 0.0786i)15-s + (−0.866 + 0.727i)17-s + (0.786 + 0.617i)19-s + (0.684 + 0.815i)21-s + (1.61 + 0.283i)23-s + (0.921 − 0.335i)25-s + (−0.0962 − 0.166i)27-s + (1.02 − 1.22i)29-s + (0.300 − 0.519i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.744 - 0.667i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.744 - 0.667i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.812494 + 0.310692i\)
\(L(\frac12)\) \(\approx\) \(0.812494 + 0.310692i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (-3.42 - 2.69i)T \)
good5 \( 1 + (0.0537 + 0.304i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (4.22 + 2.43i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.67 - 0.966i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.49 - 4.11i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (3.57 - 2.99i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-7.72 - 1.36i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-5.51 + 6.57i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (-1.67 + 2.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.17iT - 37T^{2} \)
41 \( 1 + (3.68 - 10.1i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (2.64 - 0.465i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (5.72 - 6.82i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (-11.3 - 2.00i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-2.80 + 2.35i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.0154 - 0.0878i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (3.50 + 2.93i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-2.43 - 13.8i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (11.0 + 4.02i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (4.56 + 1.66i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-9.83 - 5.67i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.29 - 11.7i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (10.0 + 11.9i)T + (-16.8 + 95.5i)T^{2} \)
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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

−10.12812575714602017344348907080, −9.577770805128388262942926417857, −8.566503490993778822769271915345, −7.40819818657310302503548020605, −6.62974468689679915186456334018, −6.19877251572031980793386091622, −4.80343356397948956230903667281, −3.95575336839438779442175998822, −2.80100207611639607274060638832, −1.06736602419780731139988290609, 0.54394667566891387571857105763, 2.87782706312635644792430712989, 3.22651344805255981960069894605, 5.03388081539103561754864227736, 5.49630675007486389300944085015, 6.68722714866545397846094563130, 7.04235138531861936031864899262, 8.737729886882930976261786338937, 8.992252568479473457459156271593, 10.18556133377100673869660161366

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