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Properties

Label 2-690-15.8-c1-0-5
Degree $2$
Conductor $690$
Sign $0.969 + 0.245i$
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 + (−1.35 − 1.07i)3-s + 1.00i·4-s + (−1.85 − 1.24i)5-s + (0.194 + 1.72i)6-s + (−2.75 + 2.75i)7-s + (0.707 − 0.707i)8-s + (0.668 + 2.92i)9-s + (0.430 + 2.19i)10-s − 2.29i·11-s + (1.07 − 1.35i)12-s + (−4.98 − 4.98i)13-s + 3.90·14-s + (1.16 + 3.69i)15-s − 1.00·16-s + (3.92 + 3.92i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.781 − 0.623i)3-s + 0.500i·4-s + (−0.829 − 0.557i)5-s + (0.0792 + 0.702i)6-s + (−1.04 + 1.04i)7-s + (0.250 − 0.250i)8-s + (0.222 + 0.974i)9-s + (0.136 + 0.693i)10-s − 0.692i·11-s + (0.311 − 0.390i)12-s + (−1.38 − 1.38i)13-s + 1.04·14-s + (0.301 + 0.953i)15-s − 0.250·16-s + (0.950 + 0.950i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.969 + 0.245i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.969 + 0.245i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.449257 - 0.0559202i\)
\(L(\frac12)\) \(\approx\) \(0.449257 - 0.0559202i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.35 + 1.07i)T \)
5 \( 1 + (1.85 + 1.24i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (2.75 - 2.75i)T - 7iT^{2} \)
11 \( 1 + 2.29iT - 11T^{2} \)
13 \( 1 + (4.98 + 4.98i)T + 13iT^{2} \)
17 \( 1 + (-3.92 - 3.92i)T + 17iT^{2} \)
19 \( 1 - 4.71iT - 19T^{2} \)
29 \( 1 - 6.12T + 29T^{2} \)
31 \( 1 - 0.118T + 31T^{2} \)
37 \( 1 + (-0.206 + 0.206i)T - 37iT^{2} \)
41 \( 1 + 5.80iT - 41T^{2} \)
43 \( 1 + (-5.81 - 5.81i)T + 43iT^{2} \)
47 \( 1 + (7.73 + 7.73i)T + 47iT^{2} \)
53 \( 1 + (-3.05 + 3.05i)T - 53iT^{2} \)
59 \( 1 - 9.92T + 59T^{2} \)
61 \( 1 - 9.71T + 61T^{2} \)
67 \( 1 + (0.772 - 0.772i)T - 67iT^{2} \)
71 \( 1 - 6.32iT - 71T^{2} \)
73 \( 1 + (0.302 + 0.302i)T + 73iT^{2} \)
79 \( 1 - 14.8iT - 79T^{2} \)
83 \( 1 + (9.99 - 9.99i)T - 83iT^{2} \)
89 \( 1 + 5.52T + 89T^{2} \)
97 \( 1 + (-1.89 + 1.89i)T - 97iT^{2} \)
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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

−10.32038092316846824315288986057, −9.850483168272735519679826919813, −8.437581226656021181620180557918, −8.093044990969155438585012460928, −7.04112516841121810983624429505, −5.85293204876849235514731761065, −5.22275719728612103832063227563, −3.63153853312277091967779175223, −2.56215104689226095739045153690, −0.802533674204954661557153710286, 0.47673105067848634361048962051, 2.97470408327127304842886817302, 4.28318496219154606077957460300, 4.87035893524804876256912182138, 6.41869023205496586948252212537, 7.09318306401074656943855758958, 7.39357557751715917396482245162, 9.048837459004224794453268755774, 9.903996945626460339844542536659, 10.13206741322655514295541305393

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