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Properties

Label 2-690-15.2-c1-0-16
Degree $2$
Conductor $690$
Sign $0.986 + 0.161i$
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 + (−1.70 + 0.292i)3-s − 1.00i·4-s + (1.73 − 1.41i)5-s + (−0.999 + 1.41i)6-s + (2.44 + 2.44i)7-s + (−0.707 − 0.707i)8-s + (2.82 − i)9-s + (0.224 − 2.22i)10-s + 5.65i·11-s + (0.292 + 1.70i)12-s + (−3.44 + 3.44i)13-s + 3.46·14-s + (−2.54 + 2.92i)15-s − 1.00·16-s + (5.19 − 5.19i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.985 + 0.169i)3-s − 0.500i·4-s + (0.774 − 0.632i)5-s + (−0.408 + 0.577i)6-s + (0.925 + 0.925i)7-s + (−0.250 − 0.250i)8-s + (0.942 − 0.333i)9-s + (0.0710 − 0.703i)10-s + 1.70i·11-s + (0.0845 + 0.492i)12-s + (−0.956 + 0.956i)13-s + 0.925·14-s + (−0.656 + 0.754i)15-s − 0.250·16-s + (1.26 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.986 + 0.161i$
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: \((2,\ 690,\ (\ :1/2),\ 0.986 + 0.161i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78947 - 0.145791i\)
\(L(\frac12)\) \(\approx\) \(1.78947 - 0.145791i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 - 0.292i)T \)
5 \( 1 + (-1.73 + 1.41i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (-2.44 - 2.44i)T + 7iT^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 + (3.44 - 3.44i)T - 13iT^{2} \)
17 \( 1 + (-5.19 + 5.19i)T - 17iT^{2} \)
19 \( 1 - 2.44iT - 19T^{2} \)
29 \( 1 - 2.19T + 29T^{2} \)
31 \( 1 - 4.89T + 31T^{2} \)
37 \( 1 + (2.44 + 2.44i)T + 37iT^{2} \)
41 \( 1 - 8.48iT - 41T^{2} \)
43 \( 1 + (-2.89 + 2.89i)T - 43iT^{2} \)
47 \( 1 + (-4.87 + 4.87i)T - 47iT^{2} \)
53 \( 1 + (-0.953 - 0.953i)T + 53iT^{2} \)
59 \( 1 - 5.51T + 59T^{2} \)
61 \( 1 - 0.898T + 61T^{2} \)
67 \( 1 + (10.8 + 10.8i)T + 67iT^{2} \)
71 \( 1 + 14.6iT - 71T^{2} \)
73 \( 1 + (1.89 - 1.89i)T - 73iT^{2} \)
79 \( 1 - 4.44iT - 79T^{2} \)
83 \( 1 + (1.41 + 1.41i)T + 83iT^{2} \)
89 \( 1 + 9.12T + 89T^{2} \)
97 \( 1 + (0.449 + 0.449i)T + 97iT^{2} \)
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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

−10.36676205679819601560152959158, −9.723925339034477010709469126027, −9.206808109308194305580122545429, −7.68002855036089286431920822470, −6.69027930406390943032652404582, −5.52160287577301073531614414162, −4.97283158503024736387380170769, −4.43861582749252643503474769174, −2.37235773137567227846136848616, −1.46267551943397379036372349319, 1.08388112414309498813505700366, 2.92749906540372938932034199923, 4.20859796228025181840564426864, 5.44230875316642904253276439154, 5.78188458946669073201116173242, 6.84066921405699762441714668959, 7.63665590752775539144992612222, 8.434485212176643862708518520309, 10.05607676255817799316206237394, 10.59268360437083933242270956404

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