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Properties

Label 1-335-335.204-r1-0-0
Degree $1$
Conductor $335$
Sign $0.374 + 0.927i$
Analytic cond. $36.0007$
Root an. cond. $36.0007$
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  + (0.841 + 0.540i)2-s + (−0.959 − 0.281i)3-s + (0.415 + 0.909i)4-s + (−0.654 − 0.755i)6-s + (0.841 + 0.540i)7-s + (−0.142 + 0.989i)8-s + (0.841 + 0.540i)9-s + (0.654 − 0.755i)11-s + (−0.142 − 0.989i)12-s + (−0.142 − 0.989i)13-s + (0.415 + 0.909i)14-s + (−0.654 + 0.755i)16-s + (−0.415 + 0.909i)17-s + (0.415 + 0.909i)18-s + (0.841 − 0.540i)19-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)2-s + (−0.959 − 0.281i)3-s + (0.415 + 0.909i)4-s + (−0.654 − 0.755i)6-s + (0.841 + 0.540i)7-s + (−0.142 + 0.989i)8-s + (0.841 + 0.540i)9-s + (0.654 − 0.755i)11-s + (−0.142 − 0.989i)12-s + (−0.142 − 0.989i)13-s + (0.415 + 0.909i)14-s + (−0.654 + 0.755i)16-s + (−0.415 + 0.909i)17-s + (0.415 + 0.909i)18-s + (0.841 − 0.540i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 335 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 335 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(335\)    =    \(5 \cdot 67\)
Sign: $0.374 + 0.927i$
Analytic conductor: \(36.0007\)
Root analytic conductor: \(36.0007\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{335} (204, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 335,\ (1:\ ),\ 0.374 + 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.359234122 + 1.591632977i\)
\(L(\frac12)\) \(\approx\) \(2.359234122 + 1.591632977i\)
\(L(1)\) \(\approx\) \(1.461648558 + 0.5485477122i\)
\(L(1)\) \(\approx\) \(1.461648558 + 0.5485477122i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.841 + 0.540i)T \)
3 \( 1 + (-0.959 - 0.281i)T \)
7 \( 1 + (0.841 + 0.540i)T \)
11 \( 1 + (0.654 - 0.755i)T \)
13 \( 1 + (-0.142 - 0.989i)T \)
17 \( 1 + (-0.415 + 0.909i)T \)
19 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (0.959 + 0.281i)T \)
29 \( 1 + T \)
31 \( 1 + (0.142 - 0.989i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.415 + 0.909i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 + (0.959 + 0.281i)T \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 + (-0.142 + 0.989i)T \)
61 \( 1 + (0.654 + 0.755i)T \)
71 \( 1 + (0.415 + 0.909i)T \)
73 \( 1 + (0.654 + 0.755i)T \)
79 \( 1 + (0.142 + 0.989i)T \)
83 \( 1 + (0.654 - 0.755i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + T \)
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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

−24.328530197419659699459506940516, −23.43794845997033315061158459612, −22.81202297919315625573203969535, −22.049427526339351364179481880549, −21.056645304225960965846297482366, −20.55106728567457265750112734522, −19.41493399537909252419315975847, −18.2871775405592730976442090798, −17.43338892807073446896026936239, −16.41900893451237036211488461499, −15.51131390953154067684053059678, −14.41818530535128974535463674924, −13.78628040584907650852164828393, −12.37522138652230107032103947718, −11.80620900481131146028469068078, −11.03083868094673978386190527880, −10.1283471516920713504200297762, −9.16836335348111781597378752388, −7.161157922876386306528377675, −6.61874494217605672994201402681, −5.097260960603080281616971584506, −4.67970824640633251073880322473, −3.62816061830537920744124215812, −1.88689684613048464914306530674, −0.87461079550286726803632413930, 1.11183859969761023919899838177, 2.65181693745787200249521577126, 4.09365617726364084562483853912, 5.20817948215320769836965241471, 5.78574442080252797672125041081, 6.82576229528933786304344618312, 7.851034069905792752083391830853, 8.81623054658272059847183126775, 10.61634494016867590857247147558, 11.44588320659842037638986267561, 12.12161654644591983064845853284, 13.09359415271175212190851124898, 13.96469226365833108583950936982, 15.164975372431116250678036238047, 15.72200369636853695783900402857, 17.02028659970476294586958928510, 17.41482747671133897616659536343, 18.38568456292620023708362401610, 19.62631371035007354644182578185, 20.87593282894630063379993346469, 21.799208352585564738211833472923, 22.2396322534911021168471016985, 23.19442924848837287260194232471, 24.17204726757181521251878770084, 24.545449311025037360905787158280

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