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Properties

Label 2-1690-5.4-c1-0-37
Degree $2$
Conductor $1690$
Sign $0.703 + 0.711i$
Analytic cond. $13.4947$
Root an. cond. $3.67351$
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  + i·2-s − 2.55i·3-s − 4-s + (−1.58 + 1.57i)5-s + 2.55·6-s + 1.86i·7-s − i·8-s − 3.51·9-s + (−1.57 − 1.58i)10-s − 4.42·11-s + 2.55i·12-s − 1.86·14-s + (4.01 + 4.05i)15-s + 16-s + 4.16i·17-s − 3.51i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.47i·3-s − 0.5·4-s + (−0.711 + 0.703i)5-s + 1.04·6-s + 0.706i·7-s − 0.353i·8-s − 1.17·9-s + (−0.497 − 0.502i)10-s − 1.33·11-s + 0.736i·12-s − 0.499·14-s + (1.03 + 1.04i)15-s + 0.250·16-s + 1.01i·17-s − 0.827i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.703 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1690\)    =    \(2 \cdot 5 \cdot 13^{2}\)
Sign: $0.703 + 0.711i$
Analytic conductor: \(13.4947\)
Root analytic conductor: \(3.67351\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1690} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1690,\ (\ :1/2),\ 0.703 + 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.032048928\)
\(L(\frac12)\) \(\approx\) \(1.032048928\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (1.58 - 1.57i)T \)
13 \( 1 \)
good3 \( 1 + 2.55iT - 3T^{2} \)
7 \( 1 - 1.86iT - 7T^{2} \)
11 \( 1 + 4.42T + 11T^{2} \)
17 \( 1 - 4.16iT - 17T^{2} \)
19 \( 1 - 4.52T + 19T^{2} \)
23 \( 1 + 7.56iT - 23T^{2} \)
29 \( 1 + 3.22T + 29T^{2} \)
31 \( 1 - 8.51T + 31T^{2} \)
37 \( 1 + 0.488iT - 37T^{2} \)
41 \( 1 - 10.9T + 41T^{2} \)
43 \( 1 + 4.38iT - 43T^{2} \)
47 \( 1 + 12.5iT - 47T^{2} \)
53 \( 1 - 0.191iT - 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 - 6.28T + 61T^{2} \)
67 \( 1 - 5.69iT - 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 - 4.10iT - 73T^{2} \)
79 \( 1 - 2.33T + 79T^{2} \)
83 \( 1 + 2.49iT - 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + 1.91iT - 97T^{2} \)
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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

−8.718294255121929469512258066446, −8.181846000693457975880910929907, −7.60184482469971276539734006128, −6.98140183965252194339644719705, −6.18032146010623467011041404236, −5.55467215506882928729778364230, −4.35140846544923080027599880106, −2.98727980854025165005434542094, −2.22567714941725294386444392202, −0.51684312798472444812812462662, 0.942969854750463373466764066242, 2.86450064540920758269247867495, 3.54546442082419377134629186505, 4.50233045315626862306813148698, 4.91205615235040276140644270342, 5.71800146621541393230747967598, 7.63357356806836437299638011624, 7.75039153769495408894472625582, 9.131198600003348777184996384696, 9.470954192391537662538556971152

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