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Properties

Label 1-2015-2015.1042-r0-0-0
Degree $1$
Conductor $2015$
Sign $-0.633 + 0.773i$
Analytic cond. $9.35762$
Root an. cond. $9.35762$
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  + (−0.978 − 0.207i)2-s + (0.207 + 0.978i)3-s + (0.913 + 0.406i)4-s − i·6-s + (0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.913 + 0.406i)9-s + (−0.587 − 0.809i)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 + (0.978 − 0.207i)18-s + (−0.951 + 0.309i)19-s + (0.743 + 0.669i)21-s + (0.406 + 0.913i)22-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (0.207 + 0.978i)3-s + (0.913 + 0.406i)4-s − i·6-s + (0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.913 + 0.406i)9-s + (−0.587 − 0.809i)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 + (0.978 − 0.207i)18-s + (−0.951 + 0.309i)19-s + (0.743 + 0.669i)21-s + (0.406 + 0.913i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.633 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.633 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.633 + 0.773i$
Analytic conductor: \(9.35762\)
Root analytic conductor: \(9.35762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (1042, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (0:\ ),\ -0.633 + 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3035373057 + 0.6405728118i\)
\(L(\frac12)\) \(\approx\) \(0.3035373057 + 0.6405728118i\)
\(L(1)\) \(\approx\) \(0.6619420218 + 0.2058056098i\)
\(L(1)\) \(\approx\) \(0.6619420218 + 0.2058056098i\)

Euler product

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

−19.66416666904184680569168465825, −18.5956601657462241931886025104, −18.31339120572886799716007010884, −17.68682590381901197630737945884, −17.098938862401042595123509355686, −16.11698764121939700023404460072, −15.18069829956138588061807962410, −14.80150388070289017825089342764, −13.920002366321431398905850840189, −12.890222572579333785349209731988, −12.22625618164711343877009012168, −11.4822755899678838040300367072, −10.8289792428114925000436759068, −9.890413388016895213189533571756, −8.81756790443822708004519240319, −8.54508903282387820628577257618, −7.64286992788262646172943775016, −7.03762565311883537159718053530, −6.30467526550481615993710300903, −5.38633638377177683363544619348, −4.50038084774190308582119259325, −2.67277017872223173347466521912, −2.401617156978123923138120096457, −1.50387747280341939059730280171, −0.341029756384013386953570959831, 1.12312652066085833427957603551, 2.18672616846369683509396745098, 3.09880073263147576456026810826, 3.9254742917481042153689676445, 4.781156273052552450821291473998, 5.814503965405413752251422677683, 6.6701373801299962462835711653, 7.92810632654399541255405139147, 8.21452630243031382625076948289, 8.97050992052659778294474342817, 9.85651197115270536195554595193, 10.64291701652609356122381688634, 10.95622791934949325617766735138, 11.63746685976136280978728851985, 12.79332091021091146869218546516, 13.691356151321883851386838068797, 14.5644967747563158690929461214, 15.320561954991806233171315900902, 15.855403296286100210376255165153, 16.81620977406355908671068419985, 17.12710935063690035514017502798, 17.91917557206032505673958957626, 18.7906123993006752622876205054, 19.66940144197458239499810376549, 20.01335910959792636125633335679

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