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Properties

Label 2-820-820.747-c1-0-121
Degree $2$
Conductor $820$
Sign $0.912 - 0.408i$
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  + (−0.363 − 1.36i)2-s − 2.89i·3-s + (−1.73 + 0.993i)4-s + (−2.03 − 0.918i)5-s + (−3.95 + 1.05i)6-s − 3.97·7-s + (1.98 + 2.01i)8-s − 5.35·9-s + (−0.514 + 3.12i)10-s + (2.11 − 2.11i)11-s + (2.87 + 5.01i)12-s − 5.49·13-s + (1.44 + 5.42i)14-s + (−2.65 + 5.89i)15-s + (2.02 − 3.44i)16-s + 4.67·17-s + ⋯
L(s)  = 1  + (−0.257 − 0.966i)2-s − 1.66i·3-s + (−0.867 + 0.496i)4-s + (−0.911 − 0.410i)5-s + (−1.61 + 0.428i)6-s − 1.50·7-s + (0.703 + 0.710i)8-s − 1.78·9-s + (−0.162 + 0.986i)10-s + (0.638 − 0.638i)11-s + (0.829 + 1.44i)12-s − 1.52·13-s + (0.385 + 1.45i)14-s + (−0.685 + 1.52i)15-s + (0.506 − 0.862i)16-s + 1.13·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.912 - 0.408i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (747, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.912 - 0.408i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.193962 + 0.0413901i\)
\(L(\frac12)\) \(\approx\) \(0.193962 + 0.0413901i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.363 + 1.36i)T \)
5 \( 1 + (2.03 + 0.918i)T \)
41 \( 1 + (3.43 - 5.40i)T \)
good3 \( 1 + 2.89iT - 3T^{2} \)
7 \( 1 + 3.97T + 7T^{2} \)
11 \( 1 + (-2.11 + 2.11i)T - 11iT^{2} \)
13 \( 1 + 5.49T + 13T^{2} \)
17 \( 1 - 4.67T + 17T^{2} \)
19 \( 1 + (-4.23 + 4.23i)T - 19iT^{2} \)
23 \( 1 + (2.64 + 2.64i)T + 23iT^{2} \)
29 \( 1 + (-5.09 + 5.09i)T - 29iT^{2} \)
31 \( 1 - 3.94iT - 31T^{2} \)
37 \( 1 + (3.93 + 3.93i)T + 37iT^{2} \)
43 \( 1 + (5.85 - 5.85i)T - 43iT^{2} \)
47 \( 1 - 7.58iT - 47T^{2} \)
53 \( 1 - 6.61T + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 - 6.04iT - 61T^{2} \)
67 \( 1 + 8.81iT - 67T^{2} \)
71 \( 1 + (7.87 - 7.87i)T - 71iT^{2} \)
73 \( 1 + (4.97 + 4.97i)T + 73iT^{2} \)
79 \( 1 + (1.04 + 1.04i)T + 79iT^{2} \)
83 \( 1 + (1.36 + 1.36i)T + 83iT^{2} \)
89 \( 1 + (-4.94 + 4.94i)T - 89iT^{2} \)
97 \( 1 - 5.20T + 97T^{2} \)
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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

−9.427865894963371899813265280906, −8.592823046807350469574161759970, −7.69283284821201394205014436524, −7.13564681029587774718477253204, −6.11423060691867695982130520778, −4.79009926574574344510199462239, −3.32327293313817073331417078450, −2.73793177766106249837457204096, −1.07874945840080566848404971779, −0.13023218930621241646673426221, 3.29765918223962123935870386954, 3.78139295328116110780545804947, 4.84529582465096214066369317401, 5.65639614121731353058936029416, 6.84764322092800903907230329595, 7.49204828863790971447001344965, 8.621654919468157842179792207883, 9.551823160860931178759765700424, 10.07858078400518538668022972764, 10.27123338888186776727944676244

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