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Properties

Label 1-2015-2015.87-r1-0-0
Degree $1$
Conductor $2015$
Sign $0.986 + 0.162i$
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 − i·3-s + (0.5 − 0.866i)4-s + (−0.5 − 0.866i)6-s + (0.866 − 0.5i)7-s − i·8-s − 9-s + (−0.5 + 0.866i)11-s + (−0.866 − 0.5i)12-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.866 + 0.5i)18-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)21-s + i·22-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s − i·3-s + (0.5 − 0.866i)4-s + (−0.5 − 0.866i)6-s + (0.866 − 0.5i)7-s − i·8-s − 9-s + (−0.5 + 0.866i)11-s + (−0.866 − 0.5i)12-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.866 + 0.5i)18-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)21-s + i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.986 + 0.162i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ 0.986 + 0.162i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.271234430 + 0.1853898318i\)
\(L(\frac12)\) \(\approx\) \(2.271234430 + 0.1853898318i\)
\(L(1)\) \(\approx\) \(1.391269220 - 0.8440469284i\)
\(L(1)\) \(\approx\) \(1.391269220 - 0.8440469284i\)

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 \( 1 - iT \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
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

−20.11619038040561746925439108828, −19.061872152531385860684751961530, −18.03492077129744487285697993329, −17.322362482408555049559066020555, −16.6216606542420459351318936798, −15.84424257336240493604176453845, −15.44953475927255802207025471199, −14.59593432480676810958936908396, −14.09824295497869299987317556012, −13.39499365673308367351064829715, −12.24535435175493150344705985417, −11.67315973901361298305991723973, −10.97440715430494363467461150542, −10.24906766830811118242223607238, −9.10816467424603621228170496722, −8.270067680477888718185323487668, −7.93666233042682004061209697989, −6.63825231534048474183112399188, −5.62429397500097478263714480785, −5.35129811225956913263732446020, −4.46921335046248205322595562805, −3.6571197491841483061893394606, −2.87987070702738530516639201486, −2.00306513843992457020261495740, −0.25581002564892470716417416269, 1.28310046480564237452569818426, 1.477147775342362418379514774950, 2.62749456109167535000022004015, 3.385670800592050370054128315528, 4.52989013226282081698984722369, 5.20137031400734341953250501873, 5.96911903930588543302618542018, 6.93063846890425610126998321333, 7.613364070235783765724191687192, 8.19154122341541174003162358893, 9.602039848524133581680337573106, 10.27646554016831885387617896697, 11.12919147467795762090347620573, 12.00788915332055133890967852974, 12.2050162296145830970529126337, 13.24463388050999514068426145951, 13.80264429303156984156088600452, 14.45797214584658244025978808578, 14.987500635112296087556258825938, 16.10235624853765118997111342878, 16.870751388887021859720554334730, 18.006738823588844473258710366541, 18.226578595189925667797291829482, 19.133001434828885418133837176460, 20.03082123325617380342001858439

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