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Properties

Label 2-820-41.37-c1-0-12
Degree $2$
Conductor $820$
Sign $-0.982 + 0.187i$
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  + 0.309·3-s + (0.309 − 0.951i)5-s + (−1.44 − 1.05i)7-s − 2.90·9-s + (−0.302 − 0.932i)11-s + (−3.05 + 2.21i)13-s + (0.0956 − 0.294i)15-s + (−0.642 − 1.97i)17-s + (−2.16 − 1.57i)19-s + (−0.448 − 0.326i)21-s + (−5.35 + 3.89i)23-s + (−0.809 − 0.587i)25-s − 1.82·27-s + (−0.261 + 0.806i)29-s + (−0.586 − 1.80i)31-s + ⋯
L(s)  = 1  + 0.178·3-s + (0.138 − 0.425i)5-s + (−0.547 − 0.398i)7-s − 0.968·9-s + (−0.0913 − 0.281i)11-s + (−0.847 + 0.615i)13-s + (0.0247 − 0.0760i)15-s + (−0.155 − 0.479i)17-s + (−0.495 − 0.360i)19-s + (−0.0979 − 0.0711i)21-s + (−1.11 + 0.811i)23-s + (−0.161 − 0.117i)25-s − 0.351·27-s + (−0.0486 + 0.149i)29-s + (−0.105 − 0.323i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.187i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.982 + 0.187i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.982 + 0.187i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0319877 - 0.337525i\)
\(L(\frac12)\) \(\approx\) \(0.0319877 - 0.337525i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.23 - 4.80i)T \)
good3 \( 1 - 0.309T + 3T^{2} \)
7 \( 1 + (1.44 + 1.05i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (0.302 + 0.932i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (3.05 - 2.21i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.642 + 1.97i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.16 + 1.57i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (5.35 - 3.89i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.261 - 0.806i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.586 + 1.80i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.0450 + 0.138i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (-2.32 + 1.68i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-5.91 + 4.29i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.81 - 5.58i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-5.41 + 3.93i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.61 + 3.35i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-1.04 + 3.21i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (4.32 + 13.3i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + 11.7T + 73T^{2} \)
79 \( 1 - 6.94T + 79T^{2} \)
83 \( 1 + 7.29T + 83T^{2} \)
89 \( 1 + (7.68 + 5.58i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (3.68 - 11.3i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.673850921626118214391422149827, −9.096434355385057115296376809816, −8.195880158240102805974856366588, −7.28924558074461229424858151085, −6.31351881582115623963298985579, −5.41678640861037669074405760929, −4.37326764908429879811443366216, −3.24709251842251734622261948129, −2.09754771828892022247759796860, −0.14721608651049345051468720271, 2.23301643024507336863306901771, 3.01241830511614420675838592715, 4.23228170178023353198260689517, 5.56112180040658334184558809992, 6.16319908950798713924881338903, 7.20712133378357648923234774946, 8.179158841701517841862410826052, 8.875605093286092102129084690545, 9.925193391830359066985965269316, 10.43505362407300303105005602293

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