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Properties

Label 2-912-19.16-c1-0-14
Degree $2$
Conductor $912$
Sign $-0.499 + 0.866i$
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 + (0.386 − 2.19i)5-s + (−0.326 − 0.565i)7-s + (0.766 − 0.642i)9-s + (−0.766 + 1.32i)11-s + (0.439 + 0.160i)13-s + (0.386 + 2.19i)15-s + (−1.61 − 1.35i)17-s + (2.23 − 3.74i)19-s + (0.5 + 0.419i)21-s + (−1.02 − 5.81i)23-s + (0.0393 + 0.0143i)25-s + (−0.500 + 0.866i)27-s + (−6.38 + 5.35i)29-s + (−4.31 − 7.48i)31-s + ⋯
L(s)  = 1  + (−0.542 + 0.197i)3-s + (0.172 − 0.980i)5-s + (−0.123 − 0.213i)7-s + (0.255 − 0.214i)9-s + (−0.230 + 0.400i)11-s + (0.121 + 0.0443i)13-s + (0.0998 + 0.566i)15-s + (−0.391 − 0.328i)17-s + (0.512 − 0.858i)19-s + (0.109 + 0.0915i)21-s + (−0.213 − 1.21i)23-s + (0.00787 + 0.00286i)25-s + (−0.0962 + 0.166i)27-s + (−1.18 + 0.994i)29-s + (−0.775 − 1.34i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.499 + 0.866i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.499 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.431779 - 0.747300i\)
\(L(\frac12)\) \(\approx\) \(0.431779 - 0.747300i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 + 3.74i)T \)
good5 \( 1 + (-0.386 + 2.19i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (0.326 + 0.565i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.766 - 1.32i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.439 - 0.160i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.61 + 1.35i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (1.02 + 5.81i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (6.38 - 5.35i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (4.31 + 7.48i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.67T + 37T^{2} \)
41 \( 1 + (3.26 - 1.18i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.78 - 10.1i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-3.55 + 2.98i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (2.07 + 11.7i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (10.9 + 9.14i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.58 + 8.98i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (0.190 - 0.160i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.772 + 4.38i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-8.54 + 3.10i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-11.3 + 4.12i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-1.85 - 3.21i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (15.7 + 5.73i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-1.89 - 1.58i)T + (16.8 + 95.5i)T^{2} \)
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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

−9.649115803841184355867982660468, −9.183308580449473791271895531955, −8.209280967820565937629615687130, −7.17615220016577166900722130630, −6.34450273569144119613925056530, −5.15243122288700441071338657322, −4.77066081655070964270984549168, −3.54527994739407467823886440158, −1.95406402843603120860328260797, −0.43176456009906107629938940263, 1.67812860904528998213661484819, 3.00106478104452798650787986273, 3.97373687612345124653041119212, 5.49409488262614678421171102448, 5.92942857052378094122328390961, 7.00578661184853364157954047780, 7.59224304584767727064381847533, 8.729879747460011539310137194073, 9.669675818211528110365241235620, 10.62004828524880075782870915194

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