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Properties

Label 2-836-19.7-c1-0-1
Degree $2$
Conductor $836$
Sign $0.910 - 0.412i$
Analytic cond. $6.67549$
Root an. cond. $2.58369$
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  + (−1.5 − 2.59i)3-s + (0.5 + 0.866i)5-s − 4·7-s + (−3 + 5.19i)9-s + 11-s + (−0.5 + 0.866i)13-s + (1.5 − 2.59i)15-s + (1.5 + 2.59i)17-s + (4 + 1.73i)19-s + (6 + 10.3i)21-s + (−0.5 + 0.866i)23-s + (2 − 3.46i)25-s + 9·27-s + (−4.5 + 7.79i)29-s + (−1.5 − 2.59i)33-s + ⋯
L(s)  = 1  + (−0.866 − 1.49i)3-s + (0.223 + 0.387i)5-s − 1.51·7-s + (−1 + 1.73i)9-s + 0.301·11-s + (−0.138 + 0.240i)13-s + (0.387 − 0.670i)15-s + (0.363 + 0.630i)17-s + (0.917 + 0.397i)19-s + (1.30 + 2.26i)21-s + (−0.104 + 0.180i)23-s + (0.400 − 0.692i)25-s + 1.73·27-s + (−0.835 + 1.44i)29-s + (−0.261 − 0.452i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(836\)    =    \(2^{2} \cdot 11 \cdot 19\)
Sign: $0.910 - 0.412i$
Analytic conductor: \(6.67549\)
Root analytic conductor: \(2.58369\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{836} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 836,\ (\ :1/2),\ 0.910 - 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.687168 + 0.148515i\)
\(L(\frac12)\) \(\approx\) \(0.687168 + 0.148515i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 - T \)
19 \( 1 + (-4 - 1.73i)T \)
good3 \( 1 + (1.5 + 2.59i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 4T + 7T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.5 + 9.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.5 - 7.79i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (5.5 - 9.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.5 - 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.33646702522366867704938640802, −9.542696485261394383766975109232, −8.422539132724706952513108316876, −7.24691163535329339986942431308, −6.86755757130513256585438832220, −6.05627277259960734746038426820, −5.44603287108929398985196088070, −3.66879317051822943924313715433, −2.50904562697765138859608183709, −1.14532152944828503315704370960, 0.44577621937465448763936199651, 2.98600152902900434212645568168, 3.81664865091574965012795658662, 4.84370855560728725893621649019, 5.67137898664213827475036895724, 6.32995388208776792573103976793, 7.47205495346441206545588922792, 9.012287357229998621241424195668, 9.611362725351919664152496586479, 9.853261417665985532128464421874

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