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Properties

Label 1-2015-2015.1382-r1-0-0
Degree $1$
Conductor $2015$
Sign $-0.917 - 0.397i$
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  + (0.207 + 0.978i)2-s + (−0.207 + 0.978i)3-s + (−0.913 + 0.406i)4-s − 6-s + (−0.587 + 0.809i)7-s + (−0.587 − 0.809i)8-s + (−0.913 − 0.406i)9-s + (0.809 + 0.587i)11-s + (−0.207 − 0.978i)12-s + (−0.913 − 0.406i)14-s + (0.669 − 0.743i)16-s + (−0.587 − 0.809i)17-s + (0.207 − 0.978i)18-s + (0.309 − 0.951i)19-s + (−0.669 − 0.743i)21-s + (−0.406 + 0.913i)22-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)2-s + (−0.207 + 0.978i)3-s + (−0.913 + 0.406i)4-s − 6-s + (−0.587 + 0.809i)7-s + (−0.587 − 0.809i)8-s + (−0.913 − 0.406i)9-s + (0.809 + 0.587i)11-s + (−0.207 − 0.978i)12-s + (−0.913 − 0.406i)14-s + (0.669 − 0.743i)16-s + (−0.587 − 0.809i)17-s + (0.207 − 0.978i)18-s + (0.309 − 0.951i)19-s + (−0.669 − 0.743i)21-s + (−0.406 + 0.913i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.917 - 0.397i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (1382, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ -0.917 - 0.397i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2750310197 + 1.328064597i\)
\(L(\frac12)\) \(\approx\) \(-0.2750310197 + 1.328064597i\)
\(L(1)\) \(\approx\) \(0.5610576017 + 0.7133274907i\)
\(L(1)\) \(\approx\) \(0.5610576017 + 0.7133274907i\)

Euler product

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

−19.3028503150429231712726507881, −18.96369083344805810321252347785, −18.00366182296005110936240150783, −17.27052969451093410417978142781, −16.8233930871975291816212948072, −15.71415703402053178884320821337, −14.40515256613713400530342252937, −14.077608696361125380976367696314, −13.25103274928702213025341187807, −12.77268467846178126499176824135, −11.97823808877184414039421732114, −11.29327370526708495588434133214, −10.63664726151794306880863438470, −9.80034316020214943439256445519, −8.90352073339559523766199934454, −8.17127127537522073536289059934, −7.210838441309405807476115892080, −6.28518166965538234908778778656, −5.7631432594655384626064581130, −4.57898580462655984682672112640, −3.64154388981276591617430368749, −3.06411093431888131783384602025, −1.840144478613246797328952322134, −1.19250222021229724918984035314, −0.34764718963989727145677561222, 0.71247749622890414900654875786, 2.61533532574672162456105831875, 3.25489029835804867335489406032, 4.503744680303080163549826792652, 4.70696937688012214065252956782, 5.75911046659959126662947808960, 6.491695186387516472346623477001, 7.05913495855395158912250098895, 8.384938144595609873968338589515, 8.99583295409306464609131137266, 9.51127079003436589745595352490, 10.21870340323348714660158583434, 11.48717378845026444326421193277, 12.02012075875735072829862420906, 12.92289772996396848695765104436, 13.78564361996842389942035306496, 14.57494694515467664039542085834, 15.33028864599256520437060425609, 15.58781014276432499073909081224, 16.51105530218011024269358345624, 16.944603114083230540394338110, 17.89239744650886772457615460069, 18.35891892952726735486695891684, 19.506039649295575331211894416215, 20.17809601079101742527178049798

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