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Properties

Label 2-690-115.68-c1-0-3
Degree $2$
Conductor $690$
Sign $0.998 - 0.0556i$
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 + (−0.707 − 0.707i)3-s − 1.00i·4-s + (−2.22 + 0.254i)5-s − 1.00·6-s + (2.54 + 2.54i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (−1.39 + 1.75i)10-s + 3.50i·11-s + (−0.707 + 0.707i)12-s + (2.40 + 2.40i)13-s + 3.60·14-s + (1.75 + 1.39i)15-s − 1.00·16-s + (1.37 + 1.37i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s − 0.500i·4-s + (−0.993 + 0.113i)5-s − 0.408·6-s + (0.962 + 0.962i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.439 + 0.553i)10-s + 1.05i·11-s + (−0.204 + 0.204i)12-s + (0.667 + 0.667i)13-s + 0.962·14-s + (0.452 + 0.359i)15-s − 0.250·16-s + (0.332 + 0.332i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.998 - 0.0556i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.998 - 0.0556i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54268 + 0.0429915i\)
\(L(\frac12)\) \(\approx\) \(1.54268 + 0.0429915i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (2.22 - 0.254i)T \)
23 \( 1 + (2.27 + 4.21i)T \)
good7 \( 1 + (-2.54 - 2.54i)T + 7iT^{2} \)
11 \( 1 - 3.50iT - 11T^{2} \)
13 \( 1 + (-2.40 - 2.40i)T + 13iT^{2} \)
17 \( 1 + (-1.37 - 1.37i)T + 17iT^{2} \)
19 \( 1 - 2.31T + 19T^{2} \)
29 \( 1 - 0.113iT - 29T^{2} \)
31 \( 1 - 1.27T + 31T^{2} \)
37 \( 1 + (-3.74 - 3.74i)T + 37iT^{2} \)
41 \( 1 - 5.08T + 41T^{2} \)
43 \( 1 + (2.71 - 2.71i)T - 43iT^{2} \)
47 \( 1 + (0.775 - 0.775i)T - 47iT^{2} \)
53 \( 1 + (2.61 - 2.61i)T - 53iT^{2} \)
59 \( 1 - 13.2iT - 59T^{2} \)
61 \( 1 + 0.841iT - 61T^{2} \)
67 \( 1 + (-5.33 - 5.33i)T + 67iT^{2} \)
71 \( 1 - 14.2T + 71T^{2} \)
73 \( 1 + (7.88 + 7.88i)T + 73iT^{2} \)
79 \( 1 - 2.37T + 79T^{2} \)
83 \( 1 + (-1.02 + 1.02i)T - 83iT^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 + (4.66 + 4.66i)T + 97iT^{2} \)
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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

−10.88808914190943522856940963839, −9.764858174050872272266980553897, −8.621696753346848115502901158350, −7.896457028709641244292400291421, −6.88649576078216111816675449190, −5.88409220493642549718577315674, −4.81802591992133692792064488905, −4.13827286483418319287239631953, −2.64332770758949114662106377787, −1.45215520441902070436073121993, 0.840993598135654946427299865818, 3.35084427591065175300400623969, 3.98485391507810729169422178758, 4.99675485052374313407652826053, 5.78132062555792403245805721640, 6.99576315845807601606183871093, 7.921697948202153187474355973592, 8.295127792316249715617198045851, 9.607802678559217304233068590181, 10.93689099556149389990531694302

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