Properties

Label 2-690-15.2-c1-0-18
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: $\chi_{690} (47, \cdot )$
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

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

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