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Properties

Label 2-820-820.103-c1-0-86
Degree $2$
Conductor $820$
Sign $0.815 + 0.578i$
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  + (−1.30 + 0.546i)2-s + 3.28·3-s + (1.40 − 1.42i)4-s + (−0.479 − 2.18i)5-s + (−4.29 + 1.79i)6-s + (−1.14 + 0.372i)7-s + (−1.05 + 2.62i)8-s + 7.82·9-s + (1.81 + 2.58i)10-s + (0.512 − 3.23i)11-s + (4.61 − 4.68i)12-s + (−2.12 − 0.691i)13-s + (1.29 − 1.11i)14-s + (−1.57 − 7.18i)15-s + (−0.0595 − 3.99i)16-s + (0.391 + 0.539i)17-s + ⋯
L(s)  = 1  + (−0.922 + 0.386i)2-s + 1.89·3-s + (0.701 − 0.712i)4-s + (−0.214 − 0.976i)5-s + (−1.75 + 0.733i)6-s + (−0.432 + 0.140i)7-s + (−0.372 + 0.928i)8-s + 2.60·9-s + (0.575 + 0.818i)10-s + (0.154 − 0.976i)11-s + (1.33 − 1.35i)12-s + (−0.590 − 0.191i)13-s + (0.344 − 0.296i)14-s + (−0.407 − 1.85i)15-s + (−0.0148 − 0.999i)16-s + (0.0949 + 0.130i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.815 + 0.578i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.815 + 0.578i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76630 - 0.563295i\)
\(L(\frac12)\) \(\approx\) \(1.76630 - 0.563295i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 - 0.546i)T \)
5 \( 1 + (0.479 + 2.18i)T \)
41 \( 1 + (-6.33 - 0.909i)T \)
good3 \( 1 - 3.28T + 3T^{2} \)
7 \( 1 + (1.14 - 0.372i)T + (5.66 - 4.11i)T^{2} \)
11 \( 1 + (-0.512 + 3.23i)T + (-10.4 - 3.39i)T^{2} \)
13 \( 1 + (2.12 + 0.691i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.391 - 0.539i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-3.18 - 1.62i)T + (11.1 + 15.3i)T^{2} \)
23 \( 1 + (1.59 + 3.12i)T + (-13.5 + 18.6i)T^{2} \)
29 \( 1 + (-10.1 + 1.60i)T + (27.5 - 8.96i)T^{2} \)
31 \( 1 + (1.82 + 2.50i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (5.35 - 0.848i)T + (35.1 - 11.4i)T^{2} \)
43 \( 1 + (0.819 - 0.417i)T + (25.2 - 34.7i)T^{2} \)
47 \( 1 + (-1.76 + 5.42i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.83 + 2.52i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.94 - 12.1i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.620 + 0.201i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (9.04 + 6.57i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (0.734 - 4.63i)T + (-67.5 - 21.9i)T^{2} \)
73 \( 1 + (-2.04 + 2.04i)T - 73iT^{2} \)
79 \( 1 + (-3.29 - 3.29i)T + 79iT^{2} \)
83 \( 1 + (7.50 - 7.50i)T - 83iT^{2} \)
89 \( 1 + (-0.0119 - 0.00610i)T + (52.3 + 72.0i)T^{2} \)
97 \( 1 + (-9.55 + 13.1i)T + (-29.9 - 92.2i)T^{2} \)
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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

−9.759340297120174693310541459823, −9.090575176719116065955737437761, −8.463047201677885023679519760032, −7.991086830639441661047614700904, −7.15524910342289102238352636406, −5.97995643592043977258637462348, −4.65159608503116890546138608098, −3.39258509522879235521335435454, −2.41902811666989639121516991946, −1.07031336007376186049406124760, 1.75278666360112562758966442297, 2.76684108395244328691283535988, 3.32724778638015025669738331044, 4.38736003071426641196183042610, 6.68444008819103155122842492934, 7.24765339072073811171293247727, 7.77645873029620784897663321205, 8.726739416499948057012288475882, 9.614119245178743654185255117645, 9.900733100809403973362366012342

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