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Properties

Label 2-690-15.2-c1-0-41
Degree $2$
Conductor $690$
Sign $-0.998 - 0.0474i$
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 + (0.911 − 1.47i)3-s − 1.00i·4-s + (−2.23 − 0.104i)5-s + (−0.396 − 1.68i)6-s + (−1.29 − 1.29i)7-s + (−0.707 − 0.707i)8-s + (−1.33 − 2.68i)9-s + (−1.65 + 1.50i)10-s − 0.254i·11-s + (−1.47 − 0.911i)12-s + (0.686 − 0.686i)13-s − 1.83·14-s + (−2.19 + 3.19i)15-s − 1.00·16-s + (−2.14 + 2.14i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.526 − 0.850i)3-s − 0.500i·4-s + (−0.998 − 0.0466i)5-s + (−0.161 − 0.688i)6-s + (−0.490 − 0.490i)7-s + (−0.250 − 0.250i)8-s + (−0.445 − 0.895i)9-s + (−0.522 + 0.476i)10-s − 0.0768i·11-s + (−0.425 − 0.263i)12-s + (0.190 − 0.190i)13-s − 0.490·14-s + (−0.565 + 0.824i)15-s − 0.250·16-s + (−0.520 + 0.520i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.998 - 0.0474i$
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: \((2,\ 690,\ (\ :1/2),\ -0.998 - 0.0474i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0319939 + 1.34884i\)
\(L(\frac12)\) \(\approx\) \(0.0319939 + 1.34884i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.911 + 1.47i)T \)
5 \( 1 + (2.23 + 0.104i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (1.29 + 1.29i)T + 7iT^{2} \)
11 \( 1 + 0.254iT - 11T^{2} \)
13 \( 1 + (-0.686 + 0.686i)T - 13iT^{2} \)
17 \( 1 + (2.14 - 2.14i)T - 17iT^{2} \)
19 \( 1 - 0.235iT - 19T^{2} \)
29 \( 1 + 0.789T + 29T^{2} \)
31 \( 1 + 2.07T + 31T^{2} \)
37 \( 1 + (4.51 + 4.51i)T + 37iT^{2} \)
41 \( 1 + 7.61iT - 41T^{2} \)
43 \( 1 + (-2.43 + 2.43i)T - 43iT^{2} \)
47 \( 1 + (-2.45 + 2.45i)T - 47iT^{2} \)
53 \( 1 + (2.31 + 2.31i)T + 53iT^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + (-1.01 - 1.01i)T + 67iT^{2} \)
71 \( 1 + 1.05iT - 71T^{2} \)
73 \( 1 + (-6.97 + 6.97i)T - 73iT^{2} \)
79 \( 1 - 12.7iT - 79T^{2} \)
83 \( 1 + (8.88 + 8.88i)T + 83iT^{2} \)
89 \( 1 - 9.41T + 89T^{2} \)
97 \( 1 + (-2.94 - 2.94i)T + 97iT^{2} \)
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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

−10.21412056239328844382790314540, −9.014637989005753768663295651984, −8.319607675645165190211045191090, −7.28172780709613454183198456342, −6.67040975508185790880988313691, −5.48306091595143284269468793474, −3.99215672723566847569906410658, −3.48331637105296232444114904879, −2.16008672505803693441657205808, −0.55189058857460421630141640512, 2.64007995392023354548444834389, 3.56342834629970379309504680247, 4.41876057563822053242477423532, 5.29257192672804103012684889400, 6.49193616133000748358333625969, 7.47141474035900430573233603428, 8.343230327699297646089276477951, 9.025911405899777579830495378625, 9.897131415312569181631959174551, 11.05552821691197642331930305739

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