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Properties

Label 2-836-11.3-c1-0-13
Degree $2$
Conductor $836$
Sign $0.0830 + 0.996i$
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  + (0.347 + 1.06i)3-s + (−1.86 − 1.35i)5-s + (−0.371 + 1.14i)7-s + (1.40 − 1.01i)9-s + (−2.22 − 2.45i)11-s + (1.27 − 0.927i)13-s + (0.802 − 2.46i)15-s + (−3.90 − 2.84i)17-s + (0.309 + 0.951i)19-s − 1.35·21-s − 5.78·23-s + (0.101 + 0.313i)25-s + (4.30 + 3.13i)27-s + (1.41 − 4.36i)29-s + (6.05 − 4.39i)31-s + ⋯
L(s)  = 1  + (0.200 + 0.617i)3-s + (−0.835 − 0.606i)5-s + (−0.140 + 0.431i)7-s + (0.467 − 0.339i)9-s + (−0.670 − 0.741i)11-s + (0.353 − 0.257i)13-s + (0.207 − 0.637i)15-s + (−0.948 − 0.688i)17-s + (0.0708 + 0.218i)19-s − 0.294·21-s − 1.20·23-s + (0.0203 + 0.0627i)25-s + (0.829 + 0.602i)27-s + (0.263 − 0.810i)29-s + (1.08 − 0.789i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0830 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0830 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(836\)    =    \(2^{2} \cdot 11 \cdot 19\)
Sign: $0.0830 + 0.996i$
Analytic conductor: \(6.67549\)
Root analytic conductor: \(2.58369\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{836} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 836,\ (\ :1/2),\ 0.0830 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.713743 - 0.656726i\)
\(L(\frac12)\) \(\approx\) \(0.713743 - 0.656726i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 + 2.45i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (-0.347 - 1.06i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (1.86 + 1.35i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.371 - 1.14i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.27 + 0.927i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.90 + 2.84i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + 5.78T + 23T^{2} \)
29 \( 1 + (-1.41 + 4.36i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-6.05 + 4.39i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.02 + 9.30i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.03 + 3.17i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.59T + 43T^{2} \)
47 \( 1 + (2.92 + 8.99i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.675 + 0.490i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.18 - 9.81i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.68 - 2.68i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 3.08T + 67T^{2} \)
71 \( 1 + (-5.18 - 3.76i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.30 + 10.1i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.27 + 3.83i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.526 + 0.382i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 0.577T + 89T^{2} \)
97 \( 1 + (6.97 - 5.06i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.968175811263273015234859057522, −9.102133075356194310531264130034, −8.386039014312782560574101579533, −7.71085220358077233196188194113, −6.46378620355648894336832769312, −5.48078369324105358678059108500, −4.39908344106559618880860825658, −3.77316330602247470928123506220, −2.50288470877091291303990347121, −0.46562717392198353132813284487, 1.62189624300568611827753423902, 2.86575675363836067577194340154, 4.05857980140044019036388767100, 4.87032925427428141862054893188, 6.55004204345543827732744035811, 6.88675889471923257657034710619, 7.943885416556339092640270028862, 8.279499173906891365176140666129, 9.749712112286933230538072683524, 10.44663955399853691336114568213

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