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Properties

Label 2-820-41.37-c1-0-3
Degree $2$
Conductor $820$
Sign $0.0708 - 0.997i$
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  − 0.435·3-s + (−0.309 + 0.951i)5-s + (0.796 + 0.578i)7-s − 2.80·9-s + (0.0342 + 0.105i)11-s + (3.64 − 2.64i)13-s + (0.134 − 0.414i)15-s + (1.66 + 5.12i)17-s + (2.64 + 1.92i)19-s + (−0.347 − 0.252i)21-s + (−6.02 + 4.37i)23-s + (−0.809 − 0.587i)25-s + 2.53·27-s + (−2.75 + 8.47i)29-s + (0.443 + 1.36i)31-s + ⋯
L(s)  = 1  − 0.251·3-s + (−0.138 + 0.425i)5-s + (0.301 + 0.218i)7-s − 0.936·9-s + (0.0103 + 0.0317i)11-s + (1.01 − 0.734i)13-s + (0.0347 − 0.107i)15-s + (0.403 + 1.24i)17-s + (0.607 + 0.441i)19-s + (−0.0757 − 0.0550i)21-s + (−1.25 + 0.912i)23-s + (−0.161 − 0.117i)25-s + 0.487·27-s + (−0.511 + 1.57i)29-s + (0.0796 + 0.245i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0708 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0708 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.0708 - 0.997i$
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: \((2,\ 820,\ (\ :1/2),\ 0.0708 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.837337 + 0.779987i\)
\(L(\frac12)\) \(\approx\) \(0.837337 + 0.779987i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.55 - 5.32i)T \)
good3 \( 1 + 0.435T + 3T^{2} \)
7 \( 1 + (-0.796 - 0.578i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (-0.0342 - 0.105i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-3.64 + 2.64i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.66 - 5.12i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.64 - 1.92i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (6.02 - 4.37i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.75 - 8.47i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.443 - 1.36i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.347 + 1.07i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (7.97 - 5.79i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-1.79 + 1.30i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.0174 + 0.0536i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-7.66 + 5.56i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-11.6 - 8.43i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (3.36 - 10.3i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-1.96 - 6.05i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + 0.130T + 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + 2.84T + 83T^{2} \)
89 \( 1 + (-8.89 - 6.46i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.463 + 1.42i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.51867481093306535759216840271, −9.737076978672502645691758472164, −8.394211852983339620953879531695, −8.187722819226738453367577286789, −6.93910355185203873225084794113, −5.81446772862806396281964453475, −5.48840972461207158091008158071, −3.86912528843345750304427748124, −3.09224866967851998754453583219, −1.52134773900055284240518474357, 0.60949201712610976936357743286, 2.27891057994979726176036151386, 3.65316159182622006641430549411, 4.66128965204715958278432093191, 5.62668725631231475323741325443, 6.43116571851381828963429976397, 7.56530465214633831189232854361, 8.372913452517986428553717008278, 9.125605695254922049054536705487, 9.983931775792467150662176301261

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