Properties

Label 2-1690-13.12-c1-0-12
Degree $2$
Conductor $1690$
Sign $-0.832 - 0.554i$
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 − 2·3-s − 4-s + i·5-s − 2i·6-s i·7-s i·8-s + 9-s − 10-s + 3i·11-s + 2·12-s + 14-s − 2i·15-s + 16-s + 6·17-s + i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.15·3-s − 0.5·4-s + 0.447i·5-s − 0.816i·6-s − 0.377i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s + 0.904i·11-s + 0.577·12-s + 0.267·14-s − 0.516i·15-s + 0.250·16-s + 1.45·17-s + 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1690\)    =    \(2 \cdot 5 \cdot 13^{2}\)
Sign: $-0.832 - 0.554i$
Analytic conductor: \(13.4947\)
Root analytic conductor: \(3.67351\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1690} (1351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1690,\ (\ :1/2),\ -0.832 - 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7416117684\)
\(L(\frac12)\) \(\approx\) \(0.7416117684\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 \)
good3 \( 1 + 2T + 3T^{2} \)
7 \( 1 + iT - 7T^{2} \)
11 \( 1 - 3iT - 11T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 5iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 - 11iT - 37T^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 - 16iT - 67T^{2} \)
71 \( 1 + 6iT - 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 - 9iT - 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−9.914426556873094708510420892655, −8.788013331691349695837513818530, −7.85520344077311721263127824338, −6.95785471849505081732997347254, −6.66367594942857843505111079507, −5.54374534945842866405808623253, −5.06829816735651235661958049337, −4.09649132276333398283785164135, −2.87190400027141439115992080595, −1.10209380582552820880046554127, 0.40828781949392869542225682420, 1.53400085738395265215912465005, 2.99821802850439519782111632727, 3.94333472943989280668535895404, 5.04399025151344942215558153199, 5.75509128570922103762031584030, 6.12706264686989985887284882143, 7.60310821297685354162782639514, 8.304668797866958594113879138169, 9.188205189321335753165777285836

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