Properties

Label 2-690-15.8-c1-0-36
Degree $2$
Conductor $690$
Sign $-0.0167 + 0.999i$
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 + (1.40 − 1.01i)3-s + 1.00i·4-s + (2.22 + 0.198i)5-s + (−1.71 − 0.275i)6-s + (0.787 − 0.787i)7-s + (0.707 − 0.707i)8-s + (0.940 − 2.84i)9-s + (−1.43 − 1.71i)10-s + 0.307i·11-s + (1.01 + 1.40i)12-s + (−4.92 − 4.92i)13-s − 1.11·14-s + (3.32 − 1.98i)15-s − 1.00·16-s + (0.790 + 0.790i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.810 − 0.585i)3-s + 0.500i·4-s + (0.996 + 0.0889i)5-s + (−0.698 − 0.112i)6-s + (0.297 − 0.297i)7-s + (0.250 − 0.250i)8-s + (0.313 − 0.949i)9-s + (−0.453 − 0.542i)10-s + 0.0926i·11-s + (0.292 + 0.405i)12-s + (−1.36 − 1.36i)13-s − 0.297·14-s + (0.859 − 0.511i)15-s − 0.250·16-s + (0.191 + 0.191i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0167 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0167 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.0167 + 0.999i$
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: \((2,\ 690,\ (\ :1/2),\ -0.0167 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25367 - 1.27482i\)
\(L(\frac12)\) \(\approx\) \(1.25367 - 1.27482i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 + 1.01i)T \)
5 \( 1 + (-2.22 - 0.198i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (-0.787 + 0.787i)T - 7iT^{2} \)
11 \( 1 - 0.307iT - 11T^{2} \)
13 \( 1 + (4.92 + 4.92i)T + 13iT^{2} \)
17 \( 1 + (-0.790 - 0.790i)T + 17iT^{2} \)
19 \( 1 - 0.691iT - 19T^{2} \)
29 \( 1 - 6.67T + 29T^{2} \)
31 \( 1 - 4.52T + 31T^{2} \)
37 \( 1 + (-1.77 + 1.77i)T - 37iT^{2} \)
41 \( 1 - 0.596iT - 41T^{2} \)
43 \( 1 + (6.70 + 6.70i)T + 43iT^{2} \)
47 \( 1 + (-6.88 - 6.88i)T + 47iT^{2} \)
53 \( 1 + (5.94 - 5.94i)T - 53iT^{2} \)
59 \( 1 - 6.38T + 59T^{2} \)
61 \( 1 + 9.95T + 61T^{2} \)
67 \( 1 + (-2.19 + 2.19i)T - 67iT^{2} \)
71 \( 1 - 4.55iT - 71T^{2} \)
73 \( 1 + (4.90 + 4.90i)T + 73iT^{2} \)
79 \( 1 - 15.7iT - 79T^{2} \)
83 \( 1 + (-6.65 + 6.65i)T - 83iT^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 + (12.8 - 12.8i)T - 97iT^{2} \)
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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

−10.05103045823422688538742128100, −9.571174788524427594915993229374, −8.510652414530118231625081522413, −7.79109839187602436648072738018, −7.01309363983914524397216879634, −5.88323843920781771312092574248, −4.62133566943057512681533894126, −3.08490975733308009575788374630, −2.37389083488768756759210251658, −1.09703947836175838073697618833, 1.82826128190255722060684597169, 2.75534433780118905964781399794, 4.53643115647868616239392165282, 5.11726803934707920524541909789, 6.40016653377770503006415888325, 7.24518597677552007303228146606, 8.355853231174019730782323316503, 8.956308634982898142915094208443, 9.867521196752695269950876945912, 10.05877681176817459998767879574

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