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Properties

Label 2-836-11.4-c1-0-5
Degree $2$
Conductor $836$
Sign $-0.979 - 0.203i$
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  + (−0.617 + 1.90i)3-s + (−2.28 + 1.66i)5-s + (1.18 + 3.66i)7-s + (−0.802 − 0.582i)9-s + (2.17 + 2.50i)11-s + (5.04 + 3.66i)13-s + (−1.74 − 5.37i)15-s + (4.02 − 2.92i)17-s + (0.309 − 0.951i)19-s − 7.69·21-s − 6.56·23-s + (0.930 − 2.86i)25-s + (−3.24 + 2.35i)27-s + (1.09 + 3.36i)29-s + (−5.51 − 4.00i)31-s + ⋯
L(s)  = 1  + (−0.356 + 1.09i)3-s + (−1.02 + 0.744i)5-s + (0.449 + 1.38i)7-s + (−0.267 − 0.194i)9-s + (0.654 + 0.755i)11-s + (1.39 + 1.01i)13-s + (−0.451 − 1.38i)15-s + (0.976 − 0.709i)17-s + (0.0708 − 0.218i)19-s − 1.67·21-s − 1.36·23-s + (0.186 − 0.572i)25-s + (−0.624 + 0.453i)27-s + (0.203 + 0.625i)29-s + (−0.990 − 0.719i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(836\)    =    \(2^{2} \cdot 11 \cdot 19\)
Sign: $-0.979 - 0.203i$
Analytic conductor: \(6.67549\)
Root analytic conductor: \(2.58369\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{836} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 836,\ (\ :1/2),\ -0.979 - 0.203i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.127706 + 1.24276i\)
\(L(\frac12)\) \(\approx\) \(0.127706 + 1.24276i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 - 2.50i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (0.617 - 1.90i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (2.28 - 1.66i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-1.18 - 3.66i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-5.04 - 3.66i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.02 + 2.92i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 + 6.56T + 23T^{2} \)
29 \( 1 + (-1.09 - 3.36i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.51 + 4.00i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.56 + 4.80i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.20 + 6.78i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + (3.05 - 9.40i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.73 + 1.98i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.09 + 6.46i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-11.1 + 8.07i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 9.47T + 67T^{2} \)
71 \( 1 + (-6.92 + 5.03i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.861 - 2.65i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.132 + 0.0961i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.32 + 4.59i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 3.81T + 89T^{2} \)
97 \( 1 + (7.28 + 5.29i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.90763602296949945390045677561, −9.577203474626703066091932198097, −9.197646673208615740397492777902, −8.106890128729307414039397526269, −7.21482826393996387400072893229, −6.10896070587908514088420723929, −5.23594800951072499953198816950, −4.12966935155127828836714584030, −3.59349845785738437811899781994, −2.03562418170673881203010207265, 0.77382887419019868365433549211, 1.30294233150956124064088061972, 3.69103949755378494985334902268, 4.03758843848881588150393590016, 5.59407452163396816265688258425, 6.37169417805091620495344184829, 7.42554188385171359547566760973, 8.094263362714658761758871766754, 8.425762487881902288800309821928, 9.983584127939525265128155106763

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