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Properties

Label 1-2015-2015.1173-r0-0-0
Degree $1$
Conductor $2015$
Sign $0.505 + 0.862i$
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  + (0.866 + 0.5i)2-s − i·3-s + (0.5 + 0.866i)4-s + (0.5 − 0.866i)6-s + (0.866 + 0.5i)7-s + i·8-s − 9-s + (0.5 + 0.866i)11-s + (0.866 − 0.5i)12-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)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 + i·22-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s − i·3-s + (0.5 + 0.866i)4-s + (0.5 − 0.866i)6-s + (0.866 + 0.5i)7-s + i·8-s − 9-s + (0.5 + 0.866i)11-s + (0.866 − 0.5i)12-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)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 + i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.505 + 0.862i$
Analytic conductor: \(9.35762\)
Root analytic conductor: \(9.35762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (1173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (0:\ ),\ 0.505 + 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.606987132 + 1.493501330i\)
\(L(\frac12)\) \(\approx\) \(2.606987132 + 1.493501330i\)
\(L(1)\) \(\approx\) \(1.818881296 + 0.4282415036i\)
\(L(1)\) \(\approx\) \(1.818881296 + 0.4282415036i\)

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 \( 1 - iT \)
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 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + iT \)
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 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
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

−20.000543699142299924434477085176, −19.44164418708549961510619069120, −18.52010567828448224298381236759, −17.47618463871985066213337262551, −16.724708648487632408206028644734, −16.09857064143309250068416514879, −15.178165092329934075315145608804, −14.66585358028490875537152720938, −14.01476724563398670760597607858, −13.42016524992880005778171272776, −12.35182810916357123057695047490, −11.4353393347148146112610389717, −11.05392101076385896425673919391, −10.451918708503339457906665539523, −9.55315493047701618575290401621, −8.75666413063079909292766962942, −7.85302353646736157256007076165, −6.678816576833733323438234143083, −5.8187484416700065900681788781, −5.17543560920322475321331677371, −4.21669616212057544212680369884, −3.882344258885790965164832426290, −2.91145482922118320620433199936, −1.90703754307256030553727449214, −0.77896031285826892621936024262, 1.29278296544603948932761339745, 2.19563244854110075179488271785, 2.84311367337703841522683204805, 4.02410287770647307479476127751, 5.07697020589241789110104714412, 5.39462457605329528898961957790, 6.66747033525181195316784112981, 7.02460590494127786557121029927, 7.75376403414609859205521073646, 8.68397686398460060303481281769, 9.22476447836967905310320841885, 10.87798550857890924326328768277, 11.53715048988847125225854417448, 12.01579729739717611405502548133, 12.807773652357348849965365084781, 13.50818771551292310919855678131, 14.10087786176800609342540019155, 15.00189983959574909162366199341, 15.28583838915349766741746991953, 16.424118644353652964320876507015, 17.35269079087754892883311872142, 17.69970151257296150155625753273, 18.374343609787584114513589733902, 19.419163683561107739640509393180, 20.26909680644221927581694371385

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