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Properties

Label 2-810-15.2-c1-0-3
Degree $2$
Conductor $810$
Sign $0.439 - 0.898i$
Analytic cond. $6.46788$
Root an. cond. $2.54320$
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−2.18 − 0.495i)5-s + (−2.74 − 2.74i)7-s + (0.707 + 0.707i)8-s + (1.89 − 1.19i)10-s + 3.97i·11-s + (−0.700 + 0.700i)13-s + 3.88·14-s − 1.00·16-s + (−0.120 + 0.120i)17-s − 1.88i·19-s + (−0.495 + 2.18i)20-s + (−2.80 − 2.80i)22-s + (3.72 + 3.72i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.975 − 0.221i)5-s + (−1.03 − 1.03i)7-s + (0.250 + 0.250i)8-s + (0.598 − 0.376i)10-s + 1.19i·11-s + (−0.194 + 0.194i)13-s + 1.03·14-s − 0.250·16-s + (−0.0291 + 0.0291i)17-s − 0.432i·19-s + (−0.110 + 0.487i)20-s + (−0.599 − 0.599i)22-s + (0.776 + 0.776i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.439 - 0.898i$
Analytic conductor: \(6.46788\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1/2),\ 0.439 - 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.602569 + 0.375991i\)
\(L(\frac12)\) \(\approx\) \(0.602569 + 0.375991i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 + (2.18 + 0.495i)T \)
good7 \( 1 + (2.74 + 2.74i)T + 7iT^{2} \)
11 \( 1 - 3.97iT - 11T^{2} \)
13 \( 1 + (0.700 - 0.700i)T - 13iT^{2} \)
17 \( 1 + (0.120 - 0.120i)T - 17iT^{2} \)
19 \( 1 + 1.88iT - 19T^{2} \)
23 \( 1 + (-3.72 - 3.72i)T + 23iT^{2} \)
29 \( 1 - 4.31T + 29T^{2} \)
31 \( 1 - 9.40T + 31T^{2} \)
37 \( 1 + (-3.26 - 3.26i)T + 37iT^{2} \)
41 \( 1 - 8.26iT - 41T^{2} \)
43 \( 1 + (-1.45 + 1.45i)T - 43iT^{2} \)
47 \( 1 + (-2.45 + 2.45i)T - 47iT^{2} \)
53 \( 1 + (3.66 + 3.66i)T + 53iT^{2} \)
59 \( 1 + 5.45T + 59T^{2} \)
61 \( 1 - 8.71T + 61T^{2} \)
67 \( 1 + (5.75 + 5.75i)T + 67iT^{2} \)
71 \( 1 - 6.94iT - 71T^{2} \)
73 \( 1 + (8.27 - 8.27i)T - 73iT^{2} \)
79 \( 1 - 13.5iT - 79T^{2} \)
83 \( 1 + (4.94 + 4.94i)T + 83iT^{2} \)
89 \( 1 - 4.87T + 89T^{2} \)
97 \( 1 + (1.05 + 1.05i)T + 97iT^{2} \)
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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

−10.06931514978325191236425024216, −9.697647061207494437991036617588, −8.624117443093859994907272764383, −7.68824937579167305107455513193, −7.04302975040357986322677610959, −6.46740259611919644834160619252, −4.88451897736290822071961891230, −4.20430060230829699414694538128, −2.95821592087078677514964756306, −0.967389410878443511087429420728, 0.56005872926030927009916394932, 2.72268537432963120114663724698, 3.21739541814227497467464076070, 4.45403683862826170145043396906, 5.86442662794818481332635282093, 6.64859056451265017857459989479, 7.76649259701336625149542124271, 8.596580621791277186806089058092, 9.094795125756275675199371145829, 10.21078678845686820698632607445

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