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Properties

Label 2-820-820.747-c1-0-58
Degree $2$
Conductor $820$
Sign $0.443 + 0.896i$
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  + (−1.23 − 0.689i)2-s − 2.13i·3-s + (1.04 + 1.70i)4-s + (1.66 + 1.49i)5-s + (−1.47 + 2.63i)6-s + 3.79·7-s + (−0.121 − 2.82i)8-s − 1.54·9-s + (−1.02 − 2.99i)10-s + (−1.18 + 1.18i)11-s + (3.63 − 2.23i)12-s − 5.25·13-s + (−4.68 − 2.61i)14-s + (3.18 − 3.55i)15-s + (−1.79 + 3.57i)16-s + 6.80·17-s + ⋯
L(s)  = 1  + (−0.873 − 0.487i)2-s − 1.23i·3-s + (0.524 + 0.851i)4-s + (0.744 + 0.667i)5-s + (−0.600 + 1.07i)6-s + 1.43·7-s + (−0.0429 − 0.999i)8-s − 0.516·9-s + (−0.325 − 0.945i)10-s + (−0.358 + 0.358i)11-s + (1.04 − 0.646i)12-s − 1.45·13-s + (−1.25 − 0.699i)14-s + (0.821 − 0.917i)15-s + (−0.449 + 0.893i)16-s + 1.64·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.443 + 0.896i$
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: \((2,\ 820,\ (\ :1/2),\ 0.443 + 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18008 - 0.732641i\)
\(L(\frac12)\) \(\approx\) \(1.18008 - 0.732641i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.23 + 0.689i)T \)
5 \( 1 + (-1.66 - 1.49i)T \)
41 \( 1 + (1.60 - 6.19i)T \)
good3 \( 1 + 2.13iT - 3T^{2} \)
7 \( 1 - 3.79T + 7T^{2} \)
11 \( 1 + (1.18 - 1.18i)T - 11iT^{2} \)
13 \( 1 + 5.25T + 13T^{2} \)
17 \( 1 - 6.80T + 17T^{2} \)
19 \( 1 + (-0.539 + 0.539i)T - 19iT^{2} \)
23 \( 1 + (-2.86 - 2.86i)T + 23iT^{2} \)
29 \( 1 + (-0.631 + 0.631i)T - 29iT^{2} \)
31 \( 1 + 6.03iT - 31T^{2} \)
37 \( 1 + (-6.56 - 6.56i)T + 37iT^{2} \)
43 \( 1 + (-6.88 + 6.88i)T - 43iT^{2} \)
47 \( 1 - 6.94iT - 47T^{2} \)
53 \( 1 + 1.29T + 53T^{2} \)
59 \( 1 + 3.32T + 59T^{2} \)
61 \( 1 + 9.00iT - 61T^{2} \)
67 \( 1 + 13.9iT - 67T^{2} \)
71 \( 1 + (4.57 - 4.57i)T - 71iT^{2} \)
73 \( 1 + (2.83 + 2.83i)T + 73iT^{2} \)
79 \( 1 + (-3.21 - 3.21i)T + 79iT^{2} \)
83 \( 1 + (3.90 + 3.90i)T + 83iT^{2} \)
89 \( 1 + (-1.76 + 1.76i)T - 89iT^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
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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

−9.970466555855784091930965412741, −9.473868519778767882178456439750, −7.959567599510104593681409486845, −7.71061182453612068100127927608, −7.06816538076185016958305716529, −5.93292868320550266272789648544, −4.76316285066813975967191509884, −2.91699963882636140325647415127, −2.07856057858761711705410511992, −1.21285849434008280053622981137, 1.21694749927357047101700774014, 2.60569537937479360974764960519, 4.47367005794038045215413577855, 5.22371718615508376482850863895, 5.60519415575547360133475747198, 7.23548220851178006165323744448, 8.008019549514862800178416926414, 8.843232132086865354842136865370, 9.522059573545434142173797165982, 10.27108203272590907000479097741

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