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Properties

Label 2-820-41.4-c1-0-3
Degree $2$
Conductor $820$
Sign $-0.832 + 0.554i$
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  + 2.67i·3-s + (−0.309 + 0.951i)5-s + (−2.42 + 3.34i)7-s − 4.15·9-s + (−0.725 + 0.235i)11-s + (−0.803 − 1.10i)13-s + (−2.54 − 0.826i)15-s + (−0.985 + 0.320i)17-s + (0.711 − 0.978i)19-s + (−8.93 − 6.49i)21-s + (5.82 − 4.23i)23-s + (−0.809 − 0.587i)25-s − 3.09i·27-s + (−2.23 − 0.726i)29-s + (1.08 + 3.35i)31-s + ⋯
L(s)  = 1  + 1.54i·3-s + (−0.138 + 0.425i)5-s + (−0.917 + 1.26i)7-s − 1.38·9-s + (−0.218 + 0.0711i)11-s + (−0.222 − 0.306i)13-s + (−0.656 − 0.213i)15-s + (−0.238 + 0.0776i)17-s + (0.163 − 0.224i)19-s + (−1.95 − 1.41i)21-s + (1.21 − 0.882i)23-s + (−0.161 − 0.117i)25-s − 0.595i·27-s + (−0.415 − 0.134i)29-s + (0.195 + 0.601i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.832 + 0.554i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.237891 - 0.785804i\)
\(L(\frac12)\) \(\approx\) \(0.237891 - 0.785804i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.27 + 3.62i)T \)
good3 \( 1 - 2.67iT - 3T^{2} \)
7 \( 1 + (2.42 - 3.34i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (0.725 - 0.235i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.803 + 1.10i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.985 - 0.320i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.711 + 0.978i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (-5.82 + 4.23i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.23 + 0.726i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.08 - 3.35i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.04 - 6.30i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (-3.77 + 2.74i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-4.65 - 6.40i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-6.89 - 2.24i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.50 - 5.45i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (6.48 + 4.70i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-5.44 - 1.76i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (11.7 - 3.83i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + 10.0T + 73T^{2} \)
79 \( 1 + 0.344iT - 79T^{2} \)
83 \( 1 + 3.90T + 83T^{2} \)
89 \( 1 + (0.376 - 0.518i)T + (-27.5 - 84.6i)T^{2} \)
97 \( 1 + (18.3 + 5.97i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.48835142942870283585288705541, −9.997255698866332579906926080478, −9.008705689082284254696430769158, −8.732729640730797240168086931572, −7.24312240871694881086358351418, −6.17106557113612863366362974894, −5.33590074112973398998330134085, −4.48016653751161013379057557599, −3.25450689141706460958844694597, −2.68039925289413038955755151634, 0.40271281508745169490745284376, 1.56190407814128136317969109121, 3.00107102949291227599938558214, 4.14441977712629248794870026603, 5.50336735817198661360388589828, 6.52279800534303473806697721769, 7.24921017461229885302654325781, 7.64118089847714845655834254979, 8.786410435034938516306649210637, 9.652916501171480767387234628238

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