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Properties

Label 1-2015-2015.1839-r1-0-0
Degree $1$
Conductor $2015$
Sign $-0.962 - 0.271i$
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  + (0.743 + 0.669i)2-s + (−0.669 − 0.743i)3-s + (0.104 + 0.994i)4-s − i·6-s + (0.587 + 0.809i)7-s + (−0.587 + 0.809i)8-s + (−0.104 + 0.994i)9-s + (−0.587 − 0.809i)11-s + (0.669 − 0.743i)12-s + (−0.104 + 0.994i)14-s + (−0.978 + 0.207i)16-s + (−0.809 − 0.587i)17-s + (−0.743 + 0.669i)18-s + (0.951 − 0.309i)19-s + (0.207 − 0.978i)21-s + (0.104 − 0.994i)22-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (−0.669 − 0.743i)3-s + (0.104 + 0.994i)4-s − i·6-s + (0.587 + 0.809i)7-s + (−0.587 + 0.809i)8-s + (−0.104 + 0.994i)9-s + (−0.587 − 0.809i)11-s + (0.669 − 0.743i)12-s + (−0.104 + 0.994i)14-s + (−0.978 + 0.207i)16-s + (−0.809 − 0.587i)17-s + (−0.743 + 0.669i)18-s + (0.951 − 0.309i)19-s + (0.207 − 0.978i)21-s + (0.104 − 0.994i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.962 - 0.271i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (1839, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ -0.962 - 0.271i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1331108441 + 0.9625675019i\)
\(L(\frac12)\) \(\approx\) \(-0.1331108441 + 0.9625675019i\)
\(L(1)\) \(\approx\) \(1.057629355 + 0.4325104027i\)
\(L(1)\) \(\approx\) \(1.057629355 + 0.4325104027i\)

Euler product

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

−19.8521447494076000966056464009, −18.51302734046640111409832066279, −17.91049485112101755508911045888, −17.29347671459955504553527156481, −16.181369538497398063780413461655, −15.76414480253801630672156410236, −14.69417449577002032495919817821, −14.45343113775262579311342839762, −13.366050772227023738813482834833, −12.61336399237462839929264789723, −11.97729572867261544928239420692, −10.99459189255268686625514584054, −10.731722951016619164295337828845, −9.96186300397437245514128892592, −9.285256988151198275525663875403, −8.07900846341625740333410376922, −6.982172441028349999096163390105, −6.268669101470572007050464266194, −5.26616890848033425984590038823, −4.67122515160616760195247296710, −4.13670169609481475434732538907, −3.23589442747693840563992952198, −2.138723356268914351544196256952, −1.116024404919000278738681586538, −0.152389611104916347986111480602, 1.13935543655094853769111842912, 2.42574074681017696833656401392, 2.932636344967411166121868500593, 4.36468840690345626906397851462, 5.12641817328791874268597654328, 5.70555540142691934131389064058, 6.33002536158032581839066898559, 7.29355971795038560584404979789, 7.92268893761747823219935774374, 8.563734930016970831928525121754, 9.58455356895619929576197220105, 11.074690446457768282743844885978, 11.44908751763976813977139134253, 12.03605379053747021764080527922, 12.98002433915880682574673504592, 13.52220537313097950533987472450, 14.120093060432354392163766926573, 15.09092875892411527397302519614, 15.980312438210812616474280655444, 16.1839963412734543982816385472, 17.41731465832060968782206019697, 17.81953025050271452862501395070, 18.4002818816341335530709465892, 19.2840770040920652454407130421, 20.23526035861877733393188248975

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