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Properties

Label 2-820-205.64-c1-0-13
Degree $2$
Conductor $820$
Sign $0.724 + 0.689i$
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  + 1.56·3-s + (1.56 − 1.60i)5-s + (0.633 − 1.95i)7-s − 0.542·9-s + (1.38 − 1.91i)11-s + (1.71 + 5.26i)13-s + (2.44 − 2.51i)15-s + (−2.66 − 1.93i)17-s + (4.64 + 1.50i)19-s + (0.993 − 3.05i)21-s + (3.85 − 1.25i)23-s + (−0.132 − 4.99i)25-s − 5.55·27-s + (−4.79 − 6.59i)29-s + (−1.10 − 0.803i)31-s + ⋯
L(s)  = 1  + 0.905·3-s + (0.697 − 0.716i)5-s + (0.239 − 0.737i)7-s − 0.180·9-s + (0.418 − 0.576i)11-s + (0.474 + 1.46i)13-s + (0.631 − 0.648i)15-s + (−0.645 − 0.469i)17-s + (1.06 + 0.346i)19-s + (0.216 − 0.667i)21-s + (0.803 − 0.261i)23-s + (−0.0264 − 0.999i)25-s − 1.06·27-s + (−0.890 − 1.22i)29-s + (−0.198 − 0.144i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.724 + 0.689i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.724 + 0.689i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21150 - 0.884026i\)
\(L(\frac12)\) \(\approx\) \(2.21150 - 0.884026i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.56 + 1.60i)T \)
41 \( 1 + (-5.74 + 2.82i)T \)
good3 \( 1 - 1.56T + 3T^{2} \)
7 \( 1 + (-0.633 + 1.95i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (-1.38 + 1.91i)T + (-3.39 - 10.4i)T^{2} \)
13 \( 1 + (-1.71 - 5.26i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.66 + 1.93i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-4.64 - 1.50i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + (-3.85 + 1.25i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (4.79 + 6.59i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.10 + 0.803i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.20 - 3.03i)T + (-11.4 + 35.1i)T^{2} \)
43 \( 1 + (7.75 - 2.52i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (-2.11 - 6.51i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (5.04 - 3.66i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.236 + 0.726i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.10 + 12.6i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-1.01 + 0.737i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (4.40 - 6.06i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 - 11.5iT - 73T^{2} \)
79 \( 1 - 7.50iT - 79T^{2} \)
83 \( 1 - 10.8iT - 83T^{2} \)
89 \( 1 + (-5.11 - 1.66i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (-7.36 + 5.35i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.690270857644286830894486081860, −9.303884097536519180733139587759, −8.588435769192553042626742927685, −7.74664721988606101054876089117, −6.69161477964391480441577483505, −5.75296472347094644653905023574, −4.56957953063326330186260476399, −3.71487133141147546328907833192, −2.40770982335630985956548480506, −1.21096392583627795973626978614, 1.77486556119779933775173823321, 2.83603905710758619532660858089, 3.51326225202268534330655161913, 5.20850903711964152091040500691, 5.85501050815424318385418878176, 7.01633274854554174651773834297, 7.81266750726770804670931398740, 8.879865886472550484546443264162, 9.232660681184338520638410310365, 10.28990640845642914437312594941

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