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Properties

Label 2-820-820.647-c1-0-8
Degree $2$
Conductor $820$
Sign $-0.637 + 0.770i$
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 − 2.89·3-s + (−1.73 + 0.993i)4-s + (2.03 + 0.918i)5-s + (−1.05 − 3.95i)6-s + 3.97i·7-s + (−1.98 − 2.01i)8-s + 5.35·9-s + (−0.514 + 3.12i)10-s + (−2.11 − 2.11i)11-s + (5.01 − 2.87i)12-s + 5.49i·13-s + (−5.42 + 1.44i)14-s + (−5.89 − 2.65i)15-s + (2.02 − 3.44i)16-s + 4.67i·17-s + ⋯
L(s)  = 1  + (0.257 + 0.966i)2-s − 1.66·3-s + (−0.867 + 0.496i)4-s + (0.911 + 0.410i)5-s + (−0.428 − 1.61i)6-s + 1.50i·7-s + (−0.703 − 0.710i)8-s + 1.78·9-s + (−0.162 + 0.986i)10-s + (−0.638 − 0.638i)11-s + (1.44 − 0.829i)12-s + 1.52i·13-s + (−1.45 + 0.385i)14-s + (−1.52 − 0.685i)15-s + (0.506 − 0.862i)16-s + 1.13i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.637 + 0.770i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.637 + 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.285755 - 0.607403i\)
\(L(\frac12)\) \(\approx\) \(0.285755 - 0.607403i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.89T + 3T^{2} \)
7 \( 1 - 3.97iT - 7T^{2} \)
11 \( 1 + (2.11 + 2.11i)T + 11iT^{2} \)
13 \( 1 - 5.49iT - 13T^{2} \)
17 \( 1 - 4.67iT - 17T^{2} \)
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 \( 1 + 7.58T + 47T^{2} \)
53 \( 1 + 6.61iT - 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 + 6.04iT - 61T^{2} \)
67 \( 1 - 8.81T + 67T^{2} \)
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 \( 1 - 5.20iT - 97T^{2} \)
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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

−10.80405204137818939205956928294, −9.878235273007713516369835898399, −9.113327548833509001794681342772, −8.128501121709414350951838355505, −6.81248248312464756013495464375, −6.23643705619147632787659017966, −5.56818954311463255767476084523, −5.23818450284954199886720417603, −3.79394959774049547557419911193, −1.98792119392067738828017524890, 0.41007926369414016226731649674, 1.32954698741867597330085417591, 3.07109579515074342973590107339, 4.58049214009721394960239349962, 5.10643391053482589043312754035, 5.74137592285831213446782251695, 6.93064081248926961468401193670, 7.82544256954630146203133816187, 9.619142057336568689025189048126, 9.858340405881037290471631325174

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