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Properties

Label 2-690-15.2-c1-0-33
Degree $2$
Conductor $690$
Sign $-0.830 + 0.557i$
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 + (−1.67 − 0.451i)3-s − 1.00i·4-s + (−0.664 − 2.13i)5-s + (−1.50 + 0.862i)6-s + (2.34 + 2.34i)7-s + (−0.707 − 0.707i)8-s + (2.59 + 1.51i)9-s + (−1.97 − 1.03i)10-s − 4.92i·11-s + (−0.451 + 1.67i)12-s + (2.78 − 2.78i)13-s + 3.31·14-s + (0.146 + 3.87i)15-s − 1.00·16-s + (−2.05 + 2.05i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.965 − 0.260i)3-s − 0.500i·4-s + (−0.297 − 0.954i)5-s + (−0.613 + 0.352i)6-s + (0.884 + 0.884i)7-s + (−0.250 − 0.250i)8-s + (0.863 + 0.503i)9-s + (−0.625 − 0.328i)10-s − 1.48i·11-s + (−0.130 + 0.482i)12-s + (0.772 − 0.772i)13-s + 0.884·14-s + (0.0377 + 0.999i)15-s − 0.250·16-s + (−0.497 + 0.497i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.830 + 0.557i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.830 + 0.557i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.373502 - 1.22630i\)
\(L(\frac12)\) \(\approx\) \(0.373502 - 1.22630i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 + 0.451i)T \)
5 \( 1 + (0.664 + 2.13i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (-2.34 - 2.34i)T + 7iT^{2} \)
11 \( 1 + 4.92iT - 11T^{2} \)
13 \( 1 + (-2.78 + 2.78i)T - 13iT^{2} \)
17 \( 1 + (2.05 - 2.05i)T - 17iT^{2} \)
19 \( 1 + 3.73iT - 19T^{2} \)
29 \( 1 + 9.53T + 29T^{2} \)
31 \( 1 + 2.15T + 31T^{2} \)
37 \( 1 + (1.67 + 1.67i)T + 37iT^{2} \)
41 \( 1 - 5.18iT - 41T^{2} \)
43 \( 1 + (-2.73 + 2.73i)T - 43iT^{2} \)
47 \( 1 + (-8.67 + 8.67i)T - 47iT^{2} \)
53 \( 1 + (-8.92 - 8.92i)T + 53iT^{2} \)
59 \( 1 + 4.44T + 59T^{2} \)
61 \( 1 + 2.83T + 61T^{2} \)
67 \( 1 + (6.60 + 6.60i)T + 67iT^{2} \)
71 \( 1 + 1.39iT - 71T^{2} \)
73 \( 1 + (-7.65 + 7.65i)T - 73iT^{2} \)
79 \( 1 + 1.62iT - 79T^{2} \)
83 \( 1 + (-0.357 - 0.357i)T + 83iT^{2} \)
89 \( 1 - 7.92T + 89T^{2} \)
97 \( 1 + (-5.52 - 5.52i)T + 97iT^{2} \)
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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

−10.67059543322276670330299125482, −9.092658289964702648443918252581, −8.553783470711423067501018821539, −7.53999368209858415103581083448, −5.94528543621356545896887083809, −5.64480354435149967900809785307, −4.75332536145491344424957182693, −3.65344714324326484959560332606, −1.95153195370058287972166836673, −0.66924739083116273308847668521, 1.85288068318129581842691527527, 3.90482433826424721616191246734, 4.27372175893122858852050085879, 5.40810974208388052821490079842, 6.45641398744305635643286001620, 7.26131881746727062241355657257, 7.63334515507784672351064079300, 9.209240674412415242401256118400, 10.21940715534401406801628083290, 10.96795558263751921221596491705

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