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Properties

Label 1-2015-2015.304-r1-0-0
Degree $1$
Conductor $2015$
Sign $0.344 + 0.938i$
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  + i·2-s + (0.5 − 0.866i)3-s − 4-s + (0.866 + 0.5i)6-s + (0.866 + 0.5i)7-s − i·8-s + (−0.5 − 0.866i)9-s + (0.866 − 0.5i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)14-s + 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 + (0.5 + 0.866i)22-s + ⋯
L(s)  = 1  + i·2-s + (0.5 − 0.866i)3-s − 4-s + (0.866 + 0.5i)6-s + (0.866 + 0.5i)7-s − i·8-s + (−0.5 − 0.866i)9-s + (0.866 − 0.5i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)14-s + 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 + (0.5 + 0.866i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.344 + 0.938i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (304, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ 0.344 + 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.471704839 + 1.725176539i\)
\(L(\frac12)\) \(\approx\) \(2.471704839 + 1.725176539i\)
\(L(1)\) \(\approx\) \(1.321367927 + 0.4039214870i\)
\(L(1)\) \(\approx\) \(1.321367927 + 0.4039214870i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
31 \( 1 \)
good2 \( 1 + iT \)
3 \( 1 + (0.5 - 0.866i)T \)
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 \( 1 + T \)
29 \( 1 + T \)
37 \( 1 + (0.866 + 0.5i)T \)
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 \( 1 + T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + iT \)
97 \( 1 + iT \)
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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

−19.86544331619273073740931811849, −19.19428523416428606185896253309, −18.15883957314472272856302336940, −17.508541025260208320210555778431, −16.90110885045990284171059555583, −15.92320508201399149509953907972, −14.978813702838698690121087718795, −14.38801189649422598931610489001, −13.78500785482608798711178746515, −13.143792215746377892192265552268, −11.85537311602743927271863111119, −11.48390893766494206827545558618, −10.68894748398008102075307503743, −10.00942837998336334901959301704, −9.20034204699558533743947912460, −8.768340252186017999648150105315, −7.78967029321581005542033438022, −6.94038689820121593445538235789, −5.3847439506421885594587510898, −4.74108991894714366427457343540, −4.224246117958495076213816647065, −3.3091758083797972206578488662, −2.5188112967595196275992876618, −1.56299195565708586328739163005, −0.587974660278501726415305775225, 0.942461502848910323867464346078, 1.52133929379517003559561400660, 2.84048206344062749156168988350, 3.752237865903676692673728903306, 4.72915061767247084363677179628, 5.660520972067294201218904816475, 6.380815552268126209384420372689, 7.017522260284603053302123477994, 7.99028352592085075177654575681, 8.43954061775886879507976933232, 9.03218126667321130201461753583, 9.91557944054492103664325196741, 11.22121727398827817011220496515, 11.93730705816881258222611259530, 12.71340672555286812328458809110, 13.556588186069153626001425854495, 14.08624679312082360217182473196, 14.88345488984029710398575350018, 15.16643033132781858119829720552, 16.32659174118789180716738021426, 17.12450029694084208043897640657, 17.670086855260839816152317455267, 18.40222197771790905103638716535, 18.91897905321560763830623175707, 19.70516666866700840204585088861

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