Properties

Label 2-690-15.8-c1-0-9
Degree $2$
Conductor $690$
Sign $0.614 - 0.789i$
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 + (−1.68 + 0.396i)3-s + 1.00i·4-s + (2.10 + 0.743i)5-s + (1.47 + 0.912i)6-s + (−2.17 + 2.17i)7-s + (0.707 − 0.707i)8-s + (2.68 − 1.33i)9-s + (−0.965 − 2.01i)10-s − 5.22i·11-s + (−0.396 − 1.68i)12-s + (2.60 + 2.60i)13-s + 3.06·14-s + (−3.85 − 0.417i)15-s − 1.00·16-s + (0.444 + 0.444i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.973 + 0.228i)3-s + 0.500i·4-s + (0.943 + 0.332i)5-s + (0.601 + 0.372i)6-s + (−0.820 + 0.820i)7-s + (0.250 − 0.250i)8-s + (0.895 − 0.445i)9-s + (−0.305 − 0.637i)10-s − 1.57i·11-s + (−0.114 − 0.486i)12-s + (0.721 + 0.721i)13-s + 0.820·14-s + (−0.994 − 0.107i)15-s − 0.250·16-s + (0.107 + 0.107i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.614 - 0.789i$
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: \((2,\ 690,\ (\ :1/2),\ 0.614 - 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.750539 + 0.366813i\)
\(L(\frac12)\) \(\approx\) \(0.750539 + 0.366813i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 - 0.396i)T \)
5 \( 1 + (-2.10 - 0.743i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (2.17 - 2.17i)T - 7iT^{2} \)
11 \( 1 + 5.22iT - 11T^{2} \)
13 \( 1 + (-2.60 - 2.60i)T + 13iT^{2} \)
17 \( 1 + (-0.444 - 0.444i)T + 17iT^{2} \)
19 \( 1 - 2.97iT - 19T^{2} \)
29 \( 1 - 1.70T + 29T^{2} \)
31 \( 1 + 5.25T + 31T^{2} \)
37 \( 1 + (2.75 - 2.75i)T - 37iT^{2} \)
41 \( 1 - 6.96iT - 41T^{2} \)
43 \( 1 + (-7.32 - 7.32i)T + 43iT^{2} \)
47 \( 1 + (-6.05 - 6.05i)T + 47iT^{2} \)
53 \( 1 + (-0.302 + 0.302i)T - 53iT^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 + 1.40T + 61T^{2} \)
67 \( 1 + (-3.64 + 3.64i)T - 67iT^{2} \)
71 \( 1 + 4.61iT - 71T^{2} \)
73 \( 1 + (-0.698 - 0.698i)T + 73iT^{2} \)
79 \( 1 - 11.7iT - 79T^{2} \)
83 \( 1 + (9.38 - 9.38i)T - 83iT^{2} \)
89 \( 1 - 3.53T + 89T^{2} \)
97 \( 1 + (13.1 - 13.1i)T - 97iT^{2} \)
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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

−10.76141720867300889997772411473, −9.677011919156943759641601677421, −9.255752016023796388980259022112, −8.264475194002301753031155879253, −6.74445752146165271263497380151, −6.07425081959228824099901900011, −5.52550746065186859554682363958, −3.87836624674373008521232736229, −2.83203419553825235870790162271, −1.30088373251562663507087269175, 0.66096825964968126462647575174, 2.05391882692771651599320456347, 4.08997709870120075375307706317, 5.22124797392750725068702183371, 5.91809102028563178231156536942, 7.02385333294255432974230667730, 7.21986839549055098425750584420, 8.736662549262607524482906659706, 9.671996080451249948539918522085, 10.29728860034637612671174736001

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