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Properties

Label 2-820-41.31-c1-0-2
Degree $2$
Conductor $820$
Sign $0.370 - 0.928i$
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  + 0.0118i·3-s + (0.309 + 0.951i)5-s + (2.84 + 3.91i)7-s + 2.99·9-s + (−0.345 − 0.112i)11-s + (−0.232 + 0.319i)13-s + (−0.0112 + 0.00366i)15-s + (−7.29 − 2.37i)17-s + (2.95 + 4.06i)19-s + (−0.0464 + 0.0337i)21-s + (−1.90 − 1.38i)23-s + (−0.809 + 0.587i)25-s + 0.0711i·27-s + (−3.30 + 1.07i)29-s + (0.840 − 2.58i)31-s + ⋯
L(s)  = 1  + 0.00684i·3-s + (0.138 + 0.425i)5-s + (1.07 + 1.48i)7-s + 0.999·9-s + (−0.104 − 0.0338i)11-s + (−0.0644 + 0.0886i)13-s + (−0.00291 + 0.000945i)15-s + (−1.76 − 0.575i)17-s + (0.678 + 0.933i)19-s + (−0.0101 + 0.00736i)21-s + (−0.397 − 0.288i)23-s + (−0.161 + 0.117i)25-s + 0.0136i·27-s + (−0.613 + 0.199i)29-s + (0.150 − 0.464i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.370 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.370 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.370 - 0.928i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.370 - 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46798 + 0.994706i\)
\(L(\frac12)\) \(\approx\) \(1.46798 + 0.994706i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.63 - 4.41i)T \)
good3 \( 1 - 0.0118iT - 3T^{2} \)
7 \( 1 + (-2.84 - 3.91i)T + (-2.16 + 6.65i)T^{2} \)
11 \( 1 + (0.345 + 0.112i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.232 - 0.319i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (7.29 + 2.37i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.95 - 4.06i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.90 + 1.38i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (3.30 - 1.07i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.840 + 2.58i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.67 - 8.22i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (0.100 + 0.0729i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-6.56 + 9.04i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.07 - 0.348i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-8.76 - 6.36i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-11.1 + 8.09i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-7.87 + 2.55i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (8.79 + 2.85i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + 10.2iT - 79T^{2} \)
83 \( 1 + 9.48T + 83T^{2} \)
89 \( 1 + (3.00 + 4.14i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (9.28 - 3.01i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.35946043700931332145925595606, −9.518333671490238025116944156770, −8.710933750787534809552599163222, −7.898713959287393830627229388538, −6.95480512877617737169578491090, −5.97844769646930982950891473949, −5.05974847276419770549052469272, −4.17224317461260667588506319773, −2.60380072969311879510699922762, −1.76458642936207906455068751013, 0.947196786589051994120581387538, 2.12086482570285744453959396675, 4.10685327122215165305978848944, 4.36944587014465687035338708013, 5.50386844429850034891199735615, 6.94564976575168948458296335027, 7.35964373086678456080134609149, 8.319001308446829919874274751436, 9.259481807126313852429821583322, 10.15786762558361649196166534503

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