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Properties

Label 1-680-680.373-r1-0-0
Degree $1$
Conductor $680$
Sign $0.850 - 0.525i$
Analytic cond. $73.0761$
Root an. cond. $73.0761$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + i·7-s − 9-s + 11-s − i·13-s + 19-s − 21-s − i·23-s − i·27-s − 29-s − 31-s + i·33-s − i·37-s + 39-s − 41-s + ⋯
L(s)  = 1  + i·3-s + i·7-s − 9-s + 11-s − i·13-s + 19-s − 21-s − i·23-s − i·27-s − 29-s − 31-s + i·33-s − i·37-s + 39-s − 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(73.0761\)
Root analytic conductor: \(73.0761\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 680,\ (1:\ ),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.380034998 - 0.3920390228i\)
\(L(\frac12)\) \(\approx\) \(1.380034998 - 0.3920390228i\)
\(L(1)\) \(\approx\) \(0.9949141922 + 0.2348673811i\)
\(L(1)\) \(\approx\) \(0.9949141922 + 0.2348673811i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good3 \( 1 + T \)
7 \( 1 + iT \)
11 \( 1 \)
13 \( 1 \)
19 \( 1 + iT \)
23 \( 1 \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 + T \)
41 \( 1 \)
43 \( 1 - iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 \)
67 \( 1 \)
71 \( 1 + T \)
73 \( 1 \)
79 \( 1 - T \)
83 \( 1 \)
89 \( 1 - iT \)
97 \( 1 \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.641832133619259682105453056171, −22.01574053463897835797891651127, −20.72623093000091758321505057331, −20.00901330459037665668279018433, −19.37601573911170779646777259518, −18.56831334749997765664013528217, −17.65603692200665708598680630359, −16.926242226649934314017758279859, −16.3392552279522043273338305189, −14.9025996117624696765418066137, −14.04673042633024639910825109970, −13.60825902330224543726069850311, −12.650435402069070716631611265905, −11.57370099730591917020538843690, −11.24521466065984211911078687795, −9.774478667731031592659859557687, −9.04580111594206204883437265724, −7.85478218589888373098046239612, −7.13261257289599617068431586377, −6.49986609023484693458306008377, −5.38508967510348833735807875038, −4.101193558865577805762371052220, −3.21557159716878377974862380655, −1.71473007269851200723126175687, −1.10577564632864073465163998534, 0.35924289258110151310687639837, 2.043323717072101795195164789425, 3.18498884762173397845028478386, 3.94741022968825102127006341843, 5.2674886998001690852459243509, 5.66977470597844179143095900414, 6.927092867138198216415308734349, 8.25636347503371679393177021623, 8.97676348256228480438506922372, 9.69147756615806694117263631577, 10.62135492111950888176888860229, 11.55925636349245190766672871187, 12.22057727579435104530978575156, 13.33030609964964375245385939247, 14.63033981450957732267515792734, 14.85893242441712966813724234867, 15.90507264875240611907277305807, 16.512805311285425266190195944714, 17.50134083675758035133075882708, 18.27884465296348465134330221766, 19.2948662888529385352830262257, 20.26071570061360544966840891836, 20.709904786597242050772408666988, 21.99312601854165693520714440279, 22.15888439780350317409653749490

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