Properties

Label 2-690-15.8-c1-0-12
Degree $2$
Conductor $690$
Sign $-0.471 - 0.881i$
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 + (−1.47 + 0.906i)3-s + 1.00i·4-s + (1.96 − 1.05i)5-s + (−1.68 − 0.402i)6-s + (−0.621 + 0.621i)7-s + (−0.707 + 0.707i)8-s + (1.35 − 2.67i)9-s + (2.14 + 0.643i)10-s + 4.12i·11-s + (−0.906 − 1.47i)12-s + (−0.734 − 0.734i)13-s − 0.878·14-s + (−1.94 + 3.34i)15-s − 1.00·16-s + (2.50 + 2.50i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.852 + 0.523i)3-s + 0.500i·4-s + (0.880 − 0.473i)5-s + (−0.687 − 0.164i)6-s + (−0.234 + 0.234i)7-s + (−0.250 + 0.250i)8-s + (0.451 − 0.892i)9-s + (0.677 + 0.203i)10-s + 1.24i·11-s + (−0.261 − 0.426i)12-s + (−0.203 − 0.203i)13-s − 0.234·14-s + (−0.502 + 0.864i)15-s − 0.250·16-s + (0.606 + 0.606i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.471 - 0.881i$
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: \((2,\ 690,\ (\ :1/2),\ -0.471 - 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.796854 + 1.32973i\)
\(L(\frac12)\) \(\approx\) \(0.796854 + 1.32973i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.47 - 0.906i)T \)
5 \( 1 + (-1.96 + 1.05i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (0.621 - 0.621i)T - 7iT^{2} \)
11 \( 1 - 4.12iT - 11T^{2} \)
13 \( 1 + (0.734 + 0.734i)T + 13iT^{2} \)
17 \( 1 + (-2.50 - 2.50i)T + 17iT^{2} \)
19 \( 1 - 7.88iT - 19T^{2} \)
29 \( 1 - 7.83T + 29T^{2} \)
31 \( 1 + 7.07T + 31T^{2} \)
37 \( 1 + (6.46 - 6.46i)T - 37iT^{2} \)
41 \( 1 + 1.77iT - 41T^{2} \)
43 \( 1 + (6.31 + 6.31i)T + 43iT^{2} \)
47 \( 1 + (-0.0280 - 0.0280i)T + 47iT^{2} \)
53 \( 1 + (9.77 - 9.77i)T - 53iT^{2} \)
59 \( 1 - 10.9T + 59T^{2} \)
61 \( 1 - 11.0T + 61T^{2} \)
67 \( 1 + (-10.8 + 10.8i)T - 67iT^{2} \)
71 \( 1 + 2.25iT - 71T^{2} \)
73 \( 1 + (8.48 + 8.48i)T + 73iT^{2} \)
79 \( 1 - 1.98iT - 79T^{2} \)
83 \( 1 + (-2.21 + 2.21i)T - 83iT^{2} \)
89 \( 1 - 6.08T + 89T^{2} \)
97 \( 1 + (-11.8 + 11.8i)T - 97iT^{2} \)
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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

−10.39245552050175573384622253932, −10.09818095698909918679616392792, −9.168724397562497504227478020337, −8.099588899174079837516141336059, −6.90016946760296620173940787962, −6.09862129506158858691459476125, −5.36398968710999862415155631777, −4.63957091974722335823485210184, −3.50043414232059009160239203704, −1.72937192908212994534291449997, 0.800451124449531447352971826237, 2.30389268321274452235139015279, 3.37034269552394963893882263645, 4.98763743474620297706964674741, 5.54208398368885154110002720535, 6.62368274806687262164813978357, 7.04718397228398093707544848797, 8.560274437566440948248646027473, 9.659029178027188098456220704570, 10.38700495829476315514363605073

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