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Properties

Label 2-912-57.50-c1-0-21
Degree $2$
Conductor $912$
Sign $0.999 + 0.0127i$
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  + (−1.63 + 0.585i)3-s + (2.85 − 1.64i)5-s + 4.86·7-s + (2.31 − 1.90i)9-s + 4.65i·11-s + (0.131 + 0.0760i)13-s + (−3.68 + 4.35i)15-s + (2.37 − 1.36i)17-s + (1.11 − 4.21i)19-s + (−7.93 + 2.84i)21-s + (−4.87 − 2.81i)23-s + (2.92 − 5.06i)25-s + (−2.65 + 4.46i)27-s + (−3.39 + 5.87i)29-s − 2.83i·31-s + ⋯
L(s)  = 1  + (−0.941 + 0.337i)3-s + (1.27 − 0.736i)5-s + 1.84·7-s + (0.771 − 0.635i)9-s + 1.40i·11-s + (0.0365 + 0.0210i)13-s + (−0.952 + 1.12i)15-s + (0.575 − 0.332i)17-s + (0.256 − 0.966i)19-s + (−1.73 + 0.621i)21-s + (−1.01 − 0.587i)23-s + (0.585 − 1.01i)25-s + (−0.511 + 0.859i)27-s + (−0.629 + 1.09i)29-s − 0.508i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0127i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.999 + 0.0127i$
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: \((2,\ 912,\ (\ :1/2),\ 0.999 + 0.0127i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82586 - 0.0116853i\)
\(L(\frac12)\) \(\approx\) \(1.82586 - 0.0116853i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.63 - 0.585i)T \)
19 \( 1 + (-1.11 + 4.21i)T \)
good5 \( 1 + (-2.85 + 1.64i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 4.86T + 7T^{2} \)
11 \( 1 - 4.65iT - 11T^{2} \)
13 \( 1 + (-0.131 - 0.0760i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.37 + 1.36i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.87 + 2.81i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.39 - 5.87i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.83iT - 31T^{2} \)
37 \( 1 - 5.14iT - 37T^{2} \)
41 \( 1 + (0.166 + 0.287i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.12 - 3.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.83 + 4.52i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.486 + 0.842i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.30 + 5.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.39 + 2.42i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.73 - 2.15i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.75 - 13.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.13 - 5.42i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.38 + 4.84i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.12iT - 83T^{2} \)
89 \( 1 + (-1.77 + 3.07i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.51 - 5.49i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.972634341089166052033913510907, −9.543245967313632379939998913558, −8.507635811372156323077789073471, −7.50058733095215649417266591553, −6.53631043299478642936757583720, −5.30401426014380588437848016013, −5.07365136212576331103576945805, −4.28490384394137148325929804757, −2.08843345725489411931889049515, −1.29053674341095909054990895651, 1.33489492353441231229555842121, 2.16228713259964207262119687382, 3.83671159599056132491741904150, 5.23918983990480754772947778038, 5.74044772093879277743587724788, 6.34926062161783233020646690549, 7.69079454527851551160593328813, 8.105773147107214845959007835473, 9.438254766759878056466803320563, 10.42427858503809016064334922528

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