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Properties

Label 1-4015-4015.1143-r0-0-0
Degree $1$
Conductor $4015$
Sign $0.552 + 0.833i$
Analytic cond. $18.6455$
Root an. cond. $18.6455$
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  + (−0.984 + 0.173i)2-s + (−0.866 + 0.5i)3-s + (0.939 − 0.342i)4-s + (0.766 − 0.642i)6-s + (−0.5 − 0.866i)7-s + (−0.866 + 0.5i)8-s + (0.5 − 0.866i)9-s + (−0.642 + 0.766i)12-s + (0.766 − 0.642i)13-s + (0.642 + 0.766i)14-s + (0.766 − 0.642i)16-s + (−0.5 + 0.866i)17-s + (−0.342 + 0.939i)18-s + (−0.173 − 0.984i)19-s + (0.866 + 0.5i)21-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (−0.866 + 0.5i)3-s + (0.939 − 0.342i)4-s + (0.766 − 0.642i)6-s + (−0.5 − 0.866i)7-s + (−0.866 + 0.5i)8-s + (0.5 − 0.866i)9-s + (−0.642 + 0.766i)12-s + (0.766 − 0.642i)13-s + (0.642 + 0.766i)14-s + (0.766 − 0.642i)16-s + (−0.5 + 0.866i)17-s + (−0.342 + 0.939i)18-s + (−0.173 − 0.984i)19-s + (0.866 + 0.5i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4015\)    =    \(5 \cdot 11 \cdot 73\)
Sign: $0.552 + 0.833i$
Analytic conductor: \(18.6455\)
Root analytic conductor: \(18.6455\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4015} (1143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4015,\ (0:\ ),\ 0.552 + 0.833i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3485389809 + 0.1870961150i\)
\(L(\frac12)\) \(\approx\) \(0.3485389809 + 0.1870961150i\)
\(L(1)\) \(\approx\) \(0.4537628758 + 0.02959194870i\)
\(L(1)\) \(\approx\) \(0.4537628758 + 0.02959194870i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
73 \( 1 \)
good2 \( 1 + (-0.984 + 0.173i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (-0.984 - 0.173i)T \)
29 \( 1 + (-0.984 + 0.173i)T \)
31 \( 1 + (0.342 - 0.939i)T \)
37 \( 1 + (-0.984 - 0.173i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.766 + 0.642i)T \)
53 \( 1 + (0.173 + 0.984i)T \)
59 \( 1 + (0.642 + 0.766i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (0.642 + 0.766i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−18.38585759330879012777807770085, −17.914528426389334846774326412179, −17.06442661064557489370046766893, −16.371527605150764851610524967564, −15.977332038351015262461976152483, −15.377579175824729706273254166069, −14.26101379872033698021003039798, −13.35320832997251299579130435914, −12.657196603696784565203042476365, −11.88581288591017635921001317775, −11.62484817813518567132697692399, −10.836400369553116880341135208994, −10.004897755187066490473081826048, −9.484482663169943805458206403721, −8.52139781404258620788442594099, −8.08732046749497951342021831482, −6.97277905908818981613939881102, −6.595196975244633556811588695952, −5.87718023599135168872364784414, −5.18016099783203650859725938868, −3.941485486708319193439725643547, −3.06461465408078600631068459054, −1.946430544642706105590623296823, −1.64655463654103466040582204498, −0.29373666160311227730080124590, 0.59375245434317043675608715538, 1.44931199100959045826473184380, 2.57191037915046734507558213431, 3.717623238970399455305620288577, 4.1679994700575629612741275045, 5.48949661392443441376958282933, 5.91301635924493126067198930703, 6.80706314286139329642504165737, 7.18016210485757440572057388359, 8.27197443853816481954533605558, 8.89208953175343668695592850634, 9.77403430950362225563423794672, 10.31659799542381877166944204422, 10.83480271954676426393244679748, 11.36201537173821225673301553660, 12.252459535233454770039252657541, 13.00863073619575361868716189175, 13.773141604729950061595461104500, 14.90006974230004257461538874816, 15.525971270982222428496964509757, 15.92509914925178425334384992539, 16.705844550900134050673998439230, 17.32728660975809196427956622622, 17.58264921580575778106824984891, 18.51269330087165201934582026642

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