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Properties

Label 1-2015-2015.149-r1-0-0
Degree $1$
Conductor $2015$
Sign $0.632 + 0.774i$
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 − 3-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)6-s + (−0.866 + 0.5i)7-s − i·8-s + 9-s + (−0.866 − 0.5i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.866 − 0.5i)18-s + (−0.866 + 0.5i)19-s + (0.866 − 0.5i)21-s − 22-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s − 3-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)6-s + (−0.866 + 0.5i)7-s − i·8-s + 9-s + (−0.866 − 0.5i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.866 − 0.5i)18-s + (−0.866 + 0.5i)19-s + (0.866 − 0.5i)21-s − 22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.632 + 0.774i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ 0.632 + 0.774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03526030984 + 0.01673164831i\)
\(L(\frac12)\) \(\approx\) \(-0.03526030984 + 0.01673164831i\)
\(L(1)\) \(\approx\) \(0.7603407944 - 0.4400754739i\)
\(L(1)\) \(\approx\) \(0.7603407944 - 0.4400754739i\)

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

−20.56789858591733113594970292477, −19.73318128319694178413681427888, −18.91755914781019511346779752906, −17.71265051120359592322852701667, −17.49682496455216906290793250795, −16.53657324898342413774835781534, −16.04776078678686898512108666499, −15.37640126995428049213771107100, −14.71018683085823052494699932007, −13.43115463293972659247450011028, −13.06136099283365604985399598409, −12.547904012711202097984050555058, −11.65157594715549351029969370698, −10.80668536701145542244387016399, −10.29198789524707825375454488233, −9.24557594342832281827460727396, −8.04509456469643591160397707884, −7.27868880072601702149229123830, −6.605631604824176928779344431054, −6.0035938544379859566770268631, −5.17722888714187660938752329691, −4.38714243453258690927826149867, −3.72675084621831363960762106770, −2.637955567013793508302755449485, −1.55267375816076612992794743415, 0.00896489700315025995634081772, 0.561085854767153456320982309, 2.02768474513121708325563022640, 2.68132450523358171362621228436, 3.793718825129918127708124144021, 4.52041315981243265066264743974, 5.48167715734422742211017675582, 5.953774482760879585097436764090, 6.6422314051773781436624012745, 7.50738227437381552967244859723, 8.8233227198697099762994131608, 9.79164952964425482275912703237, 10.42672428132267197812096343872, 11.04004852292115402413231712941, 11.90060241568880499561858211268, 12.42227704524654870545527859143, 13.16328573428796161814340731765, 13.63300636660841466714947675208, 14.83978798781955721732737983741, 15.50966577204619492641774158531, 16.21190620103220339313784110931, 16.5921299385221071102827706070, 17.87525744634925733635585845025, 18.61147164235564041381142393106, 18.98595334807629295499490708388

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