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Properties

Label 1-2015-2015.83-r1-0-0
Degree $1$
Conductor $2015$
Sign $0.353 + 0.935i$
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  + (−0.309 − 0.951i)2-s + (0.207 + 0.978i)3-s + (−0.809 + 0.587i)4-s + (0.866 − 0.5i)6-s + (0.913 + 0.406i)7-s + (0.809 + 0.587i)8-s + (−0.913 + 0.406i)9-s + (−0.994 + 0.104i)11-s + (−0.743 − 0.669i)12-s + (0.104 − 0.994i)14-s + (0.309 − 0.951i)16-s + (0.994 + 0.104i)17-s + (0.669 + 0.743i)18-s + (0.743 + 0.669i)19-s + (−0.207 + 0.978i)21-s + (0.406 + 0.913i)22-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (0.207 + 0.978i)3-s + (−0.809 + 0.587i)4-s + (0.866 − 0.5i)6-s + (0.913 + 0.406i)7-s + (0.809 + 0.587i)8-s + (−0.913 + 0.406i)9-s + (−0.994 + 0.104i)11-s + (−0.743 − 0.669i)12-s + (0.104 − 0.994i)14-s + (0.309 − 0.951i)16-s + (0.994 + 0.104i)17-s + (0.669 + 0.743i)18-s + (0.743 + 0.669i)19-s + (−0.207 + 0.978i)21-s + (0.406 + 0.913i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.353 + 0.935i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ 0.353 + 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.671193342 + 1.154941020i\)
\(L(\frac12)\) \(\approx\) \(1.671193342 + 1.154941020i\)
\(L(1)\) \(\approx\) \(1.025643384 + 0.09749851534i\)
\(L(1)\) \(\approx\) \(1.025643384 + 0.09749851534i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (0.207 + 0.978i)T \)
7 \( 1 + (0.913 + 0.406i)T \)
11 \( 1 + (-0.994 + 0.104i)T \)
17 \( 1 + (0.994 + 0.104i)T \)
19 \( 1 + (0.743 + 0.669i)T \)
23 \( 1 + (0.587 - 0.809i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.207 + 0.978i)T \)
43 \( 1 + (0.743 + 0.669i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (-0.406 - 0.913i)T \)
59 \( 1 + (0.207 + 0.978i)T \)
61 \( 1 - T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.406 - 0.913i)T \)
73 \( 1 + (-0.104 - 0.994i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (0.978 + 0.207i)T \)
89 \( 1 + (-0.587 - 0.809i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−19.22562615271584902476012569118, −18.82171515694596105002835962549, −17.99630740727509808300550552739, −17.51310948136909673397278329913, −16.961399534695593378181933323933, −15.89250343567276636429986008019, −15.290259751902072471900922577392, −14.32821721079370012100528472847, −13.92133467591371964527088742288, −13.26252882911867016782281779126, −12.451055031834449818183326509579, −11.431901020437515329774239536406, −10.71068228331990448981842972614, −9.71009113548913675682040913003, −8.87605869810333322850547157329, −8.06090002539257991162753046060, −7.48317685591201948319282136967, −7.16988529714620871164865708904, −5.861952956360259266110036405666, −5.43776147264830155139100279429, −4.5091906688553395734736662215, −3.31002780686328051745955155217, −2.202471953748167217537283505727, −1.112921972620732089328174850359, −0.51918732956410875691838847086, 0.88216200070725999814147071530, 1.944825802440323610105605462789, 2.89438488812150023092280252430, 3.4155045976275404844331565394, 4.66149211766256622792780638823, 4.967154404711246761519198930, 5.85347321103119913179742863177, 7.57757407955294370094253538762, 8.11894168380439537867543394732, 8.77721807677246874946739433286, 9.63285547252742245906622113251, 10.31902598913070539423988964630, 10.83914249872051138225854365151, 11.64720523347213925067618953374, 12.29335826357969730917162323438, 13.209833622791874324597161683091, 14.16055019022538446850849961992, 14.63833498591785027579029566568, 15.47580943103760651054549123814, 16.50902613430209914038974593231, 16.8601803254893073258395011308, 18.11770815928646801744916938614, 18.27387564353161100649023759580, 19.253667296701233930725348548725, 20.13598471514906619351306155148

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