Properties

Label 1-2015-2015.769-r1-0-0
Degree $1$
Conductor $2015$
Sign $-0.930 - 0.366i$
Analytic cond. $216.541$
Root an. cond. $216.541$
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  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s i·6-s i·7-s i·8-s + (−0.5 + 0.866i)9-s + i·11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 17-s + (0.866 − 0.5i)18-s i·19-s + (0.866 − 0.5i)21-s + (0.5 − 0.866i)22-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s i·6-s i·7-s i·8-s + (−0.5 + 0.866i)9-s + i·11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 17-s + (0.866 − 0.5i)18-s i·19-s + (0.866 − 0.5i)21-s + (0.5 − 0.866i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.930 - 0.366i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ -0.930 - 0.366i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02216754217 + 0.1167314794i\)
\(L(\frac12)\) \(\approx\) \(0.02216754217 + 0.1167314794i\)
\(L(1)\) \(\approx\) \(0.7433562288 + 0.08737959191i\)
\(L(1)\) \(\approx\) \(0.7433562288 + 0.08737959191i\)

Euler product

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

−19.10737921785734248009044960586, −18.63146133109239539380948790581, −18.15158498482151182153265094149, −17.26535534998758445055475007195, −16.45558500367901986133287490568, −15.85000667673538380425961240441, −14.84068660647517696703161277917, −14.421139836204144934313041232567, −13.69702011921581846965340686685, −12.58495670758995904485379059403, −11.97816152459877182374308229425, −11.23487388448497828634772744338, −10.21539023914919512280666372885, −9.35865626998091817635001724647, −8.72512749049498403244077584535, −8.00659385103098932876498526266, −7.6282622655722536836719144187, −6.272081149902801937383821383251, −6.099567842360600806444902385604, −5.215644699181601336240813701708, −3.61818762120042015074349889771, −2.685691739064724875613067227033, −1.92014559892431196373809108204, −1.009092697873219871540539268239, −0.02796915187059384922993864944, 1.218264347269876749677224864125, 2.137964211184647818781487178801, 3.141095378572569979711594988459, 3.82141797337340200729652868693, 4.55121765340423245448249652218, 5.58701975719457655802734874323, 7.16559571427561317418286979268, 7.32888471750774143325895553720, 8.35208525375581918661204619776, 9.13870729633054543575488410287, 9.84602869813980491400258960291, 10.32223035664757052172939797246, 10.99754854744297407286757902631, 11.83370798250572798935695074559, 12.72809977180384943759963005425, 13.59537568611220453034646422789, 14.30869841199811395574232309295, 15.30636810804475845936789117067, 15.82741731961071040679348992372, 16.72567358595278516462008597092, 17.195308327703295364585741481646, 17.884539588419623722105758300569, 18.883430136597694532733304967186, 19.66149860702406772494427421455, 20.13559411058335759886012852820

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