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Properties

Label 2-836-11.5-c1-0-14
Degree $2$
Conductor $836$
Sign $0.746 + 0.664i$
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  + (0.883 + 0.641i)3-s + (−0.280 + 0.864i)5-s + (1.85 − 1.34i)7-s + (−0.558 − 1.71i)9-s + (0.176 − 3.31i)11-s + (−1.35 − 4.16i)13-s + (−0.802 + 0.583i)15-s + (1.56 − 4.81i)17-s + (−0.809 − 0.587i)19-s + 2.50·21-s − 0.664·23-s + (3.37 + 2.45i)25-s + (1.62 − 4.99i)27-s + (−2.82 + 2.05i)29-s + (3.04 + 9.38i)31-s + ⋯
L(s)  = 1  + (0.510 + 0.370i)3-s + (−0.125 + 0.386i)5-s + (0.701 − 0.509i)7-s + (−0.186 − 0.572i)9-s + (0.0532 − 0.998i)11-s + (−0.375 − 1.15i)13-s + (−0.207 + 0.150i)15-s + (0.379 − 1.16i)17-s + (−0.185 − 0.134i)19-s + 0.546·21-s − 0.138·23-s + (0.675 + 0.490i)25-s + (0.312 − 0.960i)27-s + (−0.525 + 0.381i)29-s + (0.547 + 1.68i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.746 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(836\)    =    \(2^{2} \cdot 11 \cdot 19\)
Sign: $0.746 + 0.664i$
Analytic conductor: \(6.67549\)
Root analytic conductor: \(2.58369\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{836} (533, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 836,\ (\ :1/2),\ 0.746 + 0.664i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68855 - 0.642647i\)
\(L(\frac12)\) \(\approx\) \(1.68855 - 0.642647i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.176 + 3.31i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (-0.883 - 0.641i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.280 - 0.864i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.85 + 1.34i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.35 + 4.16i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.56 + 4.81i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 + 0.664T + 23T^{2} \)
29 \( 1 + (2.82 - 2.05i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-3.04 - 9.38i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.85 + 2.07i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.82 - 2.77i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 2.17T + 43T^{2} \)
47 \( 1 + (2.97 + 2.16i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.36 - 4.19i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.30 + 2.40i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.675 + 2.07i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 2.51T + 67T^{2} \)
71 \( 1 + (-1.39 + 4.29i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (7.56 - 5.49i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.46 - 4.50i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.98 + 12.2i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 + (-4.67 - 14.3i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.15130540636737023760121354950, −9.217823203568845951490486289117, −8.433670864715784151428578304058, −7.66242590234610811288701305249, −6.77751481367765967359165901660, −5.60896366237276067812379299908, −4.69646736032631827803451319437, −3.41582869713893047441230144288, −2.87697354202576701216866476955, −0.881960987933486226496913267336, 1.72944068160677614253607532638, 2.39581013560774322359078603059, 4.10513489637439500635042919430, 4.82539171715351075449847783262, 5.92191642900441695586949171114, 7.04194567720316447456573122739, 7.943028549552068348054234796846, 8.438260691871331001894207234195, 9.355034348029963917456878149534, 10.19396117521263277664230144768

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