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Properties

Label 1-2015-2015.428-r1-0-0
Degree $1$
Conductor $2015$
Sign $-0.931 - 0.362i$
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 + (−0.866 − 0.5i)3-s − 4-s + (0.5 − 0.866i)6-s + (−0.866 − 0.5i)7-s − i·8-s + (0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)12-s + (0.5 − 0.866i)14-s + 16-s + (0.866 + 0.5i)17-s + (−0.866 + 0.5i)18-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)21-s + (−0.866 + 0.5i)22-s + ⋯
L(s)  = 1  + i·2-s + (−0.866 − 0.5i)3-s − 4-s + (0.5 − 0.866i)6-s + (−0.866 − 0.5i)7-s − i·8-s + (0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)12-s + (0.5 − 0.866i)14-s + 16-s + (0.866 + 0.5i)17-s + (−0.866 + 0.5i)18-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)21-s + (−0.866 + 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.931 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.931 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.931 - 0.362i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (428, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ -0.931 - 0.362i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1264207855 + 0.6734014836i\)
\(L(\frac12)\) \(\approx\) \(-0.1264207855 + 0.6734014836i\)
\(L(1)\) \(\approx\) \(0.5644404809 + 0.3242338441i\)
\(L(1)\) \(\approx\) \(0.5644404809 + 0.3242338441i\)

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 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + iT \)
29 \( 1 - T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + T \)
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

−19.24652226132964590734632103810, −18.69713916319437657618792686045, −18.12913611746034391986325708026, −17.00980367762395966989724742136, −16.719901571428920311505009889380, −15.805669888550696761388272394148, −14.962822546309201229815453169, −14.107476797241224757409126260148, −13.17142067091335761857950404336, −12.5453166544732280632534827320, −11.82860636946565958395519026782, −11.289318327446380679399416120926, −10.46039600275687759957261152183, −9.8485907744668199203142526010, −9.07310025967078113228536949884, −8.54240955909587015542528051565, −7.08531360032648143630209047742, −6.18717366800674926617020192368, −5.48737696482168229673679974147, −4.72890153577520638100376848146, −3.695428856490941660676353966248, −3.20356905500520321757335459424, −2.10646109347268734636401781239, −0.77570767824912918344601280534, −0.22716580006698837685713085265, 0.92340401374054904582001727238, 1.81911156983844767842858126805, 3.53142620747645334800346595450, 4.13260084972765288142130937817, 5.17866602569557581710496087359, 5.90797305170327492502524516901, 6.508317920542222648986432933007, 7.32135995749884296850829576224, 7.71246383269357952388217064399, 8.86353699799554566566106538776, 9.940462682695857628131728221624, 10.14326457131278075599724212680, 11.38373189165155413882603258715, 12.424189659499242303554410107686, 12.7516299217998531892396933239, 13.54057974742209898380852892361, 14.36434230556537617871714111353, 15.13693965058103830528873209789, 16.0049001833125024212057852325, 16.65375645397892509175614103218, 17.1327178236753742949046845506, 17.711354635882742334979167528760, 18.62916478754542876287763414911, 19.160873345182411833919722560672, 19.86335714792888983145068868767

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