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Properties

Label 2-820-41.16-c1-0-7
Degree $2$
Conductor $820$
Sign $0.956 - 0.291i$
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  + 2.42·3-s + (−0.809 − 0.587i)5-s + (0.903 + 2.78i)7-s + 2.90·9-s + (2.41 − 1.75i)11-s + (−1.03 + 3.19i)13-s + (−1.96 − 1.42i)15-s + (4.80 − 3.49i)17-s + (0.971 + 2.98i)19-s + (2.19 + 6.75i)21-s + (1.16 − 3.57i)23-s + (0.309 + 0.951i)25-s − 0.232·27-s + (2.21 + 1.60i)29-s + (−6.26 + 4.55i)31-s + ⋯
L(s)  = 1  + 1.40·3-s + (−0.361 − 0.262i)5-s + (0.341 + 1.05i)7-s + 0.968·9-s + (0.728 − 0.529i)11-s + (−0.288 + 0.887i)13-s + (−0.507 − 0.368i)15-s + (1.16 − 0.847i)17-s + (0.222 + 0.685i)19-s + (0.479 + 1.47i)21-s + (0.242 − 0.745i)23-s + (0.0618 + 0.190i)25-s − 0.0447·27-s + (0.411 + 0.298i)29-s + (−1.12 + 0.818i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.291i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 - 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.956 - 0.291i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.956 - 0.291i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.47790 + 0.369662i\)
\(L(\frac12)\) \(\approx\) \(2.47790 + 0.369662i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.16 + 4.86i)T \)
good3 \( 1 - 2.42T + 3T^{2} \)
7 \( 1 + (-0.903 - 2.78i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-2.41 + 1.75i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.03 - 3.19i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-4.80 + 3.49i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.971 - 2.98i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-1.16 + 3.57i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-2.21 - 1.60i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (6.26 - 4.55i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-9.04 - 6.57i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (-3.10 + 9.55i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (1.85 - 5.69i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.451 + 0.327i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.46 - 4.50i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.95 + 12.1i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (6.99 + 5.07i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-5.52 + 4.01i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 + 2.77T + 79T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 + (-1.24 - 3.83i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (2.35 + 1.71i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.927089991869696034693672169151, −9.104001954917704609142383040935, −8.736426534519073977044716690528, −7.938672995204862386719960248592, −7.08892672797090449998388953021, −5.84674911762670049199497845819, −4.76601650115312526957388627130, −3.61550364621254621666570532181, −2.79081265788463479382198430233, −1.59280006140736980728185838251, 1.31400031416127515842963900554, 2.77008082425484819198428464206, 3.68105203331733564332663337591, 4.39056971708885720275926335849, 5.83660579847802453809053527138, 7.27999791116025910118207929308, 7.59821289427439570191669171617, 8.338690596645285801907305038106, 9.417194783401001781180559172353, 9.969256603351676488157330272520

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