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Properties

Label 2-820-820.659-c1-0-36
Degree $2$
Conductor $820$
Sign $0.604 - 0.796i$
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 − i)2-s + 2i·4-s + (0.707 + 2.12i)5-s + (2 − 2i)8-s + (2.12 + 2.12i)9-s + (1.41 − 2.82i)10-s + (1.29 + 0.535i)13-s − 4·16-s + (−1.12 + 0.464i)17-s − 4.24i·18-s + (−4.24 + 1.41i)20-s + (−3.99 + 3i)25-s + (−0.757 − 1.82i)26-s + (−0.0502 + 0.121i)29-s + (4 + 4i)32-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + i·4-s + (0.316 + 0.948i)5-s + (0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (0.447 − 0.894i)10-s + (0.358 + 0.148i)13-s − 16-s + (−0.271 + 0.112i)17-s − 0.999i·18-s + (−0.948 + 0.316i)20-s + (−0.799 + 0.600i)25-s + (−0.148 − 0.358i)26-s + (−0.00933 + 0.0225i)29-s + (0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.604 - 0.796i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.604 - 0.796i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.971562 + 0.482187i\)
\(L(\frac12)\) \(\approx\) \(0.971562 + 0.482187i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
5 \( 1 + (-0.707 - 2.12i)T \)
41 \( 1 + (5 - 4i)T \)
good3 \( 1 + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-1.29 - 0.535i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + (1.12 - 0.464i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (13.4 - 13.4i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (0.0502 - 0.121i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 7.07iT - 37T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (0.636 - 1.53i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (-1 + i)T - 61iT^{2} \)
67 \( 1 + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-4.24 - 4.24i)T + 73iT^{2} \)
79 \( 1 + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-5.87 + 14.1i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 + (-5.46 - 13.1i)T + (-68.5 + 68.5i)T^{2} \)
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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

−10.30464502388729714713535588597, −9.798131547579435380531427873192, −8.775555028225713965811392554667, −7.84815217648235074215299815091, −7.10586591619712400780470258334, −6.26386159139235607476504093934, −4.77628926062613873580976542477, −3.67680782512726593512284788992, −2.60899578358174830009033706252, −1.56861856044304586754879065447, 0.71776496532625049942792730332, 1.93760267105659457790815019339, 3.91642553856862501581727837160, 4.95903764263936369286490873634, 5.83431391156546928501095862156, 6.70299853214957688907033703435, 7.58615650435958078512440556933, 8.569910459179866336918164738676, 9.136042038588808443257108453444, 9.863455155609521180402633453059

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