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Properties

Label 2-836-11.5-c1-0-2
Degree $2$
Conductor $836$
Sign $-0.605 - 0.795i$
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  + (−1.16 − 0.848i)3-s + (−1.32 + 4.08i)5-s + (2.57 − 1.87i)7-s + (−0.283 − 0.872i)9-s + (−0.809 + 3.21i)11-s + (0.275 + 0.848i)13-s + (5.01 − 3.64i)15-s + (−0.0463 + 0.142i)17-s + (−0.809 − 0.587i)19-s − 4.59·21-s + 1.71·23-s + (−10.8 − 7.89i)25-s + (−1.74 + 5.37i)27-s + (−5.22 + 3.79i)29-s + (1.86 + 5.74i)31-s + ⋯
L(s)  = 1  + (−0.674 − 0.489i)3-s + (−0.593 + 1.82i)5-s + (0.973 − 0.707i)7-s + (−0.0945 − 0.290i)9-s + (−0.243 + 0.969i)11-s + (0.0764 + 0.235i)13-s + (1.29 − 0.940i)15-s + (−0.0112 + 0.0345i)17-s + (−0.185 − 0.134i)19-s − 1.00·21-s + 0.356·23-s + (−2.17 − 1.57i)25-s + (−0.336 + 1.03i)27-s + (−0.969 + 0.704i)29-s + (0.335 + 1.03i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(836\)    =    \(2^{2} \cdot 11 \cdot 19\)
Sign: $-0.605 - 0.795i$
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: \((2,\ 836,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.294326 + 0.593692i\)
\(L(\frac12)\) \(\approx\) \(0.294326 + 0.593692i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 - 3.21i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (1.16 + 0.848i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (1.32 - 4.08i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-2.57 + 1.87i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.275 - 0.848i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.0463 - 0.142i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 - 1.71T + 23T^{2} \)
29 \( 1 + (5.22 - 3.79i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.86 - 5.74i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.78 - 3.47i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.38 + 4.63i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 9.58T + 43T^{2} \)
47 \( 1 + (-3.17 - 2.30i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.76 - 11.5i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.74 + 1.26i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.889 + 2.73i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 0.0354T + 67T^{2} \)
71 \( 1 + (4.84 - 14.9i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.16 - 2.29i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.789 - 2.42i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.380 + 1.17i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 0.565T + 89T^{2} \)
97 \( 1 + (1.08 + 3.34i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.62140430877248928640248164108, −10.09748440242049539537193958204, −8.657798683080725511551993676795, −7.49731542058445402955213807481, −7.09238385000204310967555732731, −6.52614234132947986937429551276, −5.26138489882371405397044045414, −4.11060483285159314372542688169, −3.09869115483831248314023201442, −1.67828441378672123691493092143, 0.35364957760752139279940225567, 1.88506371712207558101134904247, 3.76317294363545918120800914771, 4.83017916474974646057462795231, 5.24630414921289254307231841503, 5.94233949950729879360605866695, 7.74806415019534484559215440772, 8.346263584363790362199354850689, 8.826474238028886716190374165860, 9.883331039602109011928678843916

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