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Properties

Label 1-2015-2015.457-r1-0-0
Degree $1$
Conductor $2015$
Sign $-0.0483 - 0.998i$
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  + (0.913 − 0.406i)2-s + (0.994 − 0.104i)3-s + (0.669 − 0.743i)4-s + (0.866 − 0.5i)6-s + (−0.669 + 0.743i)7-s + (0.309 − 0.951i)8-s + (0.978 − 0.207i)9-s + (−0.207 + 0.978i)11-s + (0.587 − 0.809i)12-s + (−0.309 + 0.951i)14-s + (−0.104 − 0.994i)16-s + (−0.207 − 0.978i)17-s + (0.809 − 0.587i)18-s + (−0.994 − 0.104i)19-s + (−0.587 + 0.809i)21-s + (0.207 + 0.978i)22-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)2-s + (0.994 − 0.104i)3-s + (0.669 − 0.743i)4-s + (0.866 − 0.5i)6-s + (−0.669 + 0.743i)7-s + (0.309 − 0.951i)8-s + (0.978 − 0.207i)9-s + (−0.207 + 0.978i)11-s + (0.587 − 0.809i)12-s + (−0.309 + 0.951i)14-s + (−0.104 − 0.994i)16-s + (−0.207 − 0.978i)17-s + (0.809 − 0.587i)18-s + (−0.994 − 0.104i)19-s + (−0.587 + 0.809i)21-s + (0.207 + 0.978i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0483 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0483 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.0483 - 0.998i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ -0.0483 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.875341569 - 4.067470960i\)
\(L(\frac12)\) \(\approx\) \(3.875341569 - 4.067470960i\)
\(L(1)\) \(\approx\) \(2.259679609 - 0.8576636216i\)
\(L(1)\) \(\approx\) \(2.259679609 - 0.8576636216i\)

Euler product

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

−19.7320189476556632829512547459, −19.6418962101243858911761939446, −18.75246080368443627089290088203, −17.50032401919883354717642978556, −16.80432337150789767773144000780, −16.07277402681536439192408456907, −15.50574466191498177133452422559, −14.71059468706292077541702251537, −14.07324413556039914834051986483, −13.363935992581060231582396220945, −12.97087436310560196018749473284, −12.18468587792906221775519753771, −10.800895415186243390761609032781, −10.60090565235154027311476547776, −9.32287216107108071030951916046, −8.50809590468845766472998020206, −7.89916527167834185649235581040, −6.99619428883498398955343472716, −6.4023078954299635100192153614, −5.45037239543245687371576799342, −4.332767072167048290312125705339, −3.771786993182275427367584689039, −3.09779088406359580678897700951, −2.27762167914499600172623648229, −1.08010427870136796632884428641, 0.57952527290661378131149950024, 1.90254497009060298740092367717, 2.53608500778286436910439230064, 3.066008151020620065216292491493, 4.1755635628730621502955895033, 4.750239902089594464732897306550, 5.77448243601213660140620260056, 6.856384628466276511936826991551, 7.12573118365687489887204804120, 8.47900642694697097292363652831, 9.15010331676946741122829786865, 9.99386694905974122780626478336, 10.51424741710388760247189372175, 11.79371187191836882680993294555, 12.359567243331562449427981198696, 13.00779407289821471719120477068, 13.58019463232159500034395837421, 14.44939044337572711398256630214, 15.11002435457465621953257920573, 15.57910748298996433983007208755, 16.24001003028410387986814225553, 17.49013898122297774844963799279, 18.56951192492115722570035575123, 18.985626717070863469546802001, 19.67253502260202173614541077403

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