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Properties

Label 2-690-115.22-c1-0-17
Degree $2$
Conductor $690$
Sign $0.892 - 0.451i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
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 + (−0.707 + 0.707i)3-s + 1.00i·4-s + (1.57 − 1.59i)5-s − 1.00·6-s + (1.37 − 1.37i)7-s + (−0.707 + 0.707i)8-s − 1.00i·9-s + (2.23 − 0.0144i)10-s − 0.0288i·11-s + (−0.707 − 0.707i)12-s + (0.711 − 0.711i)13-s + 1.95·14-s + (0.0144 + 2.23i)15-s − 1.00·16-s + (0.438 − 0.438i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.408 + 0.408i)3-s + 0.500i·4-s + (0.702 − 0.711i)5-s − 0.408·6-s + (0.521 − 0.521i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (0.707 − 0.00456i)10-s − 0.00870i·11-s + (−0.204 − 0.204i)12-s + (0.197 − 0.197i)13-s + 0.521·14-s + (0.00372 + 0.577i)15-s − 0.250·16-s + (0.106 − 0.106i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.892 - 0.451i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.892 - 0.451i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02541 + 0.483262i\)
\(L(\frac12)\) \(\approx\) \(2.02541 + 0.483262i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-1.57 + 1.59i)T \)
23 \( 1 + (-3.68 + 3.06i)T \)
good7 \( 1 + (-1.37 + 1.37i)T - 7iT^{2} \)
11 \( 1 + 0.0288iT - 11T^{2} \)
13 \( 1 + (-0.711 + 0.711i)T - 13iT^{2} \)
17 \( 1 + (-0.438 + 0.438i)T - 17iT^{2} \)
19 \( 1 - 7.23T + 19T^{2} \)
29 \( 1 - 3.97iT - 29T^{2} \)
31 \( 1 + 5.20T + 31T^{2} \)
37 \( 1 + (1.46 - 1.46i)T - 37iT^{2} \)
41 \( 1 + 7.17T + 41T^{2} \)
43 \( 1 + (-5.83 - 5.83i)T + 43iT^{2} \)
47 \( 1 + (-6.90 - 6.90i)T + 47iT^{2} \)
53 \( 1 + (2.73 + 2.73i)T + 53iT^{2} \)
59 \( 1 - 5.73iT - 59T^{2} \)
61 \( 1 + 5.09iT - 61T^{2} \)
67 \( 1 + (-4.63 + 4.63i)T - 67iT^{2} \)
71 \( 1 + 5.47T + 71T^{2} \)
73 \( 1 + (0.908 - 0.908i)T - 73iT^{2} \)
79 \( 1 - 5.20T + 79T^{2} \)
83 \( 1 + (9.95 + 9.95i)T + 83iT^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 + (-0.850 + 0.850i)T - 97iT^{2} \)
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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

−10.58742134387126455210505749747, −9.585275282255904692235798938831, −8.866525007584765042709779623866, −7.81511663050487094276575636452, −6.91486757325982433166590274875, −5.78726024617245615756689481432, −5.13925685124778591011794863543, −4.38630894509382391523324053273, −3.09753964176372934797529101888, −1.23531347112690269907780207092, 1.46016851189311194934655008507, 2.53184930059703392869980498508, 3.68609343851086248583489959229, 5.30871671715894530825303533957, 5.58397744144485872684748103924, 6.78466648329175886933041396497, 7.53269327773137077103991556714, 8.883532955732172334194933114793, 9.730052518741534037139767702827, 10.59079074360092500961396646391

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