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Properties

Label 2-820-20.7-c1-0-62
Degree $2$
Conductor $820$
Sign $-0.414 + 0.910i$
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  + (−1.35 − 0.412i)2-s + (0.638 + 0.638i)3-s + (1.66 + 1.11i)4-s + (−0.848 + 2.06i)5-s + (−0.600 − 1.12i)6-s + (−1.94 + 1.94i)7-s + (−1.78 − 2.19i)8-s − 2.18i·9-s + (2.00 − 2.44i)10-s − 4.03i·11-s + (0.347 + 1.77i)12-s + (−2.07 + 2.07i)13-s + (3.44 − 1.83i)14-s + (−1.86 + 0.778i)15-s + (1.51 + 3.70i)16-s + (−5.74 − 5.74i)17-s + ⋯
L(s)  = 1  + (−0.956 − 0.291i)2-s + (0.368 + 0.368i)3-s + (0.830 + 0.557i)4-s + (−0.379 + 0.925i)5-s + (−0.244 − 0.459i)6-s + (−0.736 + 0.736i)7-s + (−0.631 − 0.775i)8-s − 0.728i·9-s + (0.632 − 0.774i)10-s − 1.21i·11-s + (0.100 + 0.511i)12-s + (−0.574 + 0.574i)13-s + (0.919 − 0.490i)14-s + (−0.480 + 0.201i)15-s + (0.377 + 0.925i)16-s + (−1.39 − 1.39i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.414 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.414 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.414 + 0.910i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.414 + 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.171190 - 0.266038i\)
\(L(\frac12)\) \(\approx\) \(0.171190 - 0.266038i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 + 0.412i)T \)
5 \( 1 + (0.848 - 2.06i)T \)
41 \( 1 - T \)
good3 \( 1 + (-0.638 - 0.638i)T + 3iT^{2} \)
7 \( 1 + (1.94 - 1.94i)T - 7iT^{2} \)
11 \( 1 + 4.03iT - 11T^{2} \)
13 \( 1 + (2.07 - 2.07i)T - 13iT^{2} \)
17 \( 1 + (5.74 + 5.74i)T + 17iT^{2} \)
19 \( 1 + 2.15T + 19T^{2} \)
23 \( 1 + (-5.64 - 5.64i)T + 23iT^{2} \)
29 \( 1 + 3.18iT - 29T^{2} \)
31 \( 1 + 6.99iT - 31T^{2} \)
37 \( 1 + (0.454 + 0.454i)T + 37iT^{2} \)
43 \( 1 + (-0.276 - 0.276i)T + 43iT^{2} \)
47 \( 1 + (-4.84 + 4.84i)T - 47iT^{2} \)
53 \( 1 + (1.74 - 1.74i)T - 53iT^{2} \)
59 \( 1 - 10.6T + 59T^{2} \)
61 \( 1 + 15.0T + 61T^{2} \)
67 \( 1 + (1.79 - 1.79i)T - 67iT^{2} \)
71 \( 1 + 1.40iT - 71T^{2} \)
73 \( 1 + (-2.55 + 2.55i)T - 73iT^{2} \)
79 \( 1 + 6.18T + 79T^{2} \)
83 \( 1 + (3.37 + 3.37i)T + 83iT^{2} \)
89 \( 1 + 10.6iT - 89T^{2} \)
97 \( 1 + (9.91 + 9.91i)T + 97iT^{2} \)
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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

−9.652960778960450423512697101567, −9.255794508690534072131129642470, −8.603824439511910965906576226308, −7.39974255029266011222043130881, −6.69853254582146193654984606210, −5.93486002584224294681402087588, −4.09071617289096135496064502859, −3.05156686102047549873447356548, −2.53212846220720591327004334706, −0.19822592302452492424095418304, 1.46192463173466593101754146499, 2.62348774757637905838574818259, 4.29214929052455608362681231586, 5.18252543748108125126143140876, 6.67427444266074393805959988567, 7.12640923417326587732001937727, 8.092633527750427950893502803366, 8.673138413325020478703372428423, 9.485419415119947142368031044494, 10.57693168208291851554148507088

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