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Properties

Label 2-820-820.647-c1-0-69
Degree $2$
Conductor $820$
Sign $0.987 + 0.159i$
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  + (−1.36 − 0.363i)2-s + 2.89·3-s + (1.73 + 0.993i)4-s + (2.03 + 0.918i)5-s + (−3.95 − 1.05i)6-s − 3.97i·7-s + (−2.01 − 1.98i)8-s + 5.35·9-s + (−2.45 − 1.99i)10-s + (2.11 + 2.11i)11-s + (5.01 + 2.87i)12-s + 5.49i·13-s + (−1.44 + 5.42i)14-s + (5.89 + 2.65i)15-s + (2.02 + 3.44i)16-s + 4.67i·17-s + ⋯
L(s)  = 1  + (−0.966 − 0.257i)2-s + 1.66·3-s + (0.867 + 0.496i)4-s + (0.911 + 0.410i)5-s + (−1.61 − 0.428i)6-s − 1.50i·7-s + (−0.710 − 0.703i)8-s + 1.78·9-s + (−0.775 − 0.631i)10-s + (0.638 + 0.638i)11-s + (1.44 + 0.829i)12-s + 1.52i·13-s + (−0.385 + 1.45i)14-s + (1.52 + 0.685i)15-s + (0.506 + 0.862i)16-s + 1.13i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.987 + 0.159i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.987 + 0.159i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13147 - 0.171298i\)
\(L(\frac12)\) \(\approx\) \(2.13147 - 0.171298i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.363i)T \)
5 \( 1 + (-2.03 - 0.918i)T \)
41 \( 1 + (3.43 + 5.40i)T \)
good3 \( 1 - 2.89T + 3T^{2} \)
7 \( 1 + 3.97iT - 7T^{2} \)
11 \( 1 + (-2.11 - 2.11i)T + 11iT^{2} \)
13 \( 1 - 5.49iT - 13T^{2} \)
17 \( 1 - 4.67iT - 17T^{2} \)
19 \( 1 + (4.23 + 4.23i)T + 19iT^{2} \)
23 \( 1 + (-2.64 - 2.64i)T + 23iT^{2} \)
29 \( 1 + (5.09 + 5.09i)T + 29iT^{2} \)
31 \( 1 + 3.94iT - 31T^{2} \)
37 \( 1 + (3.93 + 3.93i)T + 37iT^{2} \)
43 \( 1 + (5.85 - 5.85i)T - 43iT^{2} \)
47 \( 1 - 7.58T + 47T^{2} \)
53 \( 1 + 6.61iT - 53T^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 + 6.04iT - 61T^{2} \)
67 \( 1 + 8.81T + 67T^{2} \)
71 \( 1 + (7.87 + 7.87i)T + 71iT^{2} \)
73 \( 1 + (-4.97 - 4.97i)T + 73iT^{2} \)
79 \( 1 + (-1.04 + 1.04i)T - 79iT^{2} \)
83 \( 1 + (-1.36 - 1.36i)T + 83iT^{2} \)
89 \( 1 + (4.94 + 4.94i)T + 89iT^{2} \)
97 \( 1 - 5.20iT - 97T^{2} \)
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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

−9.973409728714131480745176529579, −9.279447693533504294937860865740, −8.837808469187750876284719334401, −7.69994155376899420754304902299, −7.02164218739702933040663624205, −6.50101573587516653855822378587, −4.20548813252853670763695993938, −3.62170770314345184756496614151, −2.18729947629363250848021375777, −1.65342465768356427571828918287, 1.49848178725229381037836397661, 2.54477095369799700338690767465, 3.18366252961165102870345164216, 5.20853301850136771082022902565, 5.95840949240079090170271298267, 7.05877931107237253718951417962, 8.205752613621150007516464557322, 8.768239247882958527019121614429, 8.983158809404379486131384315034, 9.909491281355468675265895020818

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