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Properties

Label 2-820-41.25-c1-0-1
Degree $2$
Conductor $820$
Sign $0.327 - 0.944i$
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  − 2.63i·3-s + (−0.809 − 0.587i)5-s + (−4.34 + 1.41i)7-s − 3.96·9-s + (3.69 + 5.08i)11-s + (−4.60 − 1.49i)13-s + (−1.55 + 2.13i)15-s + (3.25 + 4.48i)17-s + (0.300 − 0.0976i)19-s + (3.72 + 11.4i)21-s + (0.230 − 0.709i)23-s + (0.309 + 0.951i)25-s + 2.53i·27-s + (−3.51 + 4.83i)29-s + (2.31 − 1.67i)31-s + ⋯
L(s)  = 1  − 1.52i·3-s + (−0.361 − 0.262i)5-s + (−1.64 + 0.533i)7-s − 1.32·9-s + (1.11 + 1.53i)11-s + (−1.27 − 0.415i)13-s + (−0.400 + 0.551i)15-s + (0.790 + 1.08i)17-s + (0.0689 − 0.0224i)19-s + (0.813 + 2.50i)21-s + (0.0480 − 0.147i)23-s + (0.0618 + 0.190i)25-s + 0.487i·27-s + (−0.652 + 0.898i)29-s + (0.415 − 0.301i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.327 - 0.944i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (681, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.327 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.375522 + 0.267216i\)
\(L(\frac12)\) \(\approx\) \(0.375522 + 0.267216i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (6.33 + 0.957i)T \)
good3 \( 1 + 2.63iT - 3T^{2} \)
7 \( 1 + (4.34 - 1.41i)T + (5.66 - 4.11i)T^{2} \)
11 \( 1 + (-3.69 - 5.08i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (4.60 + 1.49i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-3.25 - 4.48i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.300 + 0.0976i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.230 + 0.709i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (3.51 - 4.83i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.31 + 1.67i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (4.76 + 3.45i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (3.10 - 9.54i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-0.0516 - 0.0167i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.66 + 2.29i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.273 + 0.842i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.43 + 7.49i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (7.54 - 10.3i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-9.19 - 12.6i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + 5.83T + 73T^{2} \)
79 \( 1 - 14.5iT - 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + (-3.44 + 1.11i)T + (72.0 - 52.3i)T^{2} \)
97 \( 1 + (-0.509 + 0.700i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.10170814637862741444183133191, −9.603430669933383413312596261427, −8.617137529850089423211572802769, −7.58036082460220257377676705093, −6.94162321304926088928335610812, −6.38798708314843944720185434103, −5.30476724781401933112024015001, −3.83985057584351671318436171420, −2.65763050086858373544233378358, −1.51155102212964306605123959034, 0.22677515265992723748297548637, 3.14386425716606058254910344673, 3.45399315221015935228330338910, 4.44490928847274514796543609614, 5.56453512261261378922277758358, 6.54874653288469345316005386240, 7.37236417722429393444040388223, 8.786788861041033158147209213927, 9.431294366552554828665547346628, 9.978345597856984182917069378784

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