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Properties

Label 2-820-41.10-c1-0-7
Degree $2$
Conductor $820$
Sign $0.687 - 0.725i$
Analytic cond. $6.54773$
Root an. cond. $2.55885$
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  + 3.07·3-s + (0.309 + 0.951i)5-s + (−2.25 + 1.63i)7-s + 6.47·9-s + (−1.30 + 4.02i)11-s + (−0.246 − 0.178i)13-s + (0.951 + 2.92i)15-s + (−1.50 + 4.63i)17-s + (6.63 − 4.82i)19-s + (−6.93 + 5.03i)21-s + (1.47 + 1.06i)23-s + (−0.809 + 0.587i)25-s + 10.6·27-s + (−0.727 − 2.24i)29-s + (1.34 − 4.14i)31-s + ⋯
L(s)  = 1  + 1.77·3-s + (0.138 + 0.425i)5-s + (−0.851 + 0.618i)7-s + 2.15·9-s + (−0.394 + 1.21i)11-s + (−0.0682 − 0.0495i)13-s + (0.245 + 0.755i)15-s + (−0.365 + 1.12i)17-s + (1.52 − 1.10i)19-s + (−1.51 + 1.09i)21-s + (0.307 + 0.223i)23-s + (−0.161 + 0.117i)25-s + 2.05·27-s + (−0.135 − 0.415i)29-s + (0.241 − 0.743i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 - 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.687 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.687 - 0.725i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.687 - 0.725i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.43944 + 1.04904i\)
\(L(\frac12)\) \(\approx\) \(2.43944 + 1.04904i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (-6.32 + 0.970i)T \)
good3 \( 1 - 3.07T + 3T^{2} \)
7 \( 1 + (2.25 - 1.63i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (1.30 - 4.02i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.246 + 0.178i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.50 - 4.63i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-6.63 + 4.82i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.47 - 1.06i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.727 + 2.24i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.34 + 4.14i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.44 + 4.45i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-3.39 - 2.46i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (2.72 + 1.98i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.92 + 9.00i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.93 + 5.76i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.89 + 1.37i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.11 + 6.49i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.55 + 4.77i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 - 9.74T + 83T^{2} \)
89 \( 1 + (-4.05 + 2.94i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-0.935 - 2.87i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.814604662186883634084957908168, −9.558418644318366400989895514886, −8.801542834920713927046591901464, −7.72523153987666760373456731146, −7.23273207359176632679515622253, −6.16175936194213779744640808662, −4.74785830232832800125313158163, −3.59398208950966675722159136402, −2.76847601060861898236912496360, −1.99535890122837388140602912382, 1.18386070066949823807858964983, 2.88526784044924845976136915801, 3.28866275250974482082649966536, 4.42434162325170325413598857379, 5.71731929359609159352517578325, 7.02046077948126353764776227343, 7.67408176388516746030539659122, 8.528965745215320038113509577100, 9.230002318315436452102145466831, 9.819854932840820420337124195655

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