Properties

Label 1-2015-2015.779-r0-0-0
Degree $1$
Conductor $2015$
Sign $0.962 + 0.272i$
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  + (−0.809 + 0.587i)2-s + (0.809 + 0.587i)3-s + (0.309 − 0.951i)4-s − 6-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (0.309 + 0.951i)9-s + (−0.309 + 0.951i)11-s + (0.809 − 0.587i)12-s + (0.309 + 0.951i)14-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)17-s + (−0.809 − 0.587i)18-s + (0.809 − 0.587i)19-s + (0.809 − 0.587i)21-s + (−0.309 − 0.951i)22-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.809 + 0.587i)3-s + (0.309 − 0.951i)4-s − 6-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (0.309 + 0.951i)9-s + (−0.309 + 0.951i)11-s + (0.809 − 0.587i)12-s + (0.309 + 0.951i)14-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)17-s + (−0.809 − 0.587i)18-s + (0.809 − 0.587i)19-s + (0.809 − 0.587i)21-s + (−0.309 − 0.951i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.962 + 0.272i$
Analytic conductor: \(9.35762\)
Root analytic conductor: \(9.35762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (779, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (0:\ ),\ 0.962 + 0.272i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.449776769 + 0.2016622011i\)
\(L(\frac12)\) \(\approx\) \(1.449776769 + 0.2016622011i\)
\(L(1)\) \(\approx\) \(0.9712885611 + 0.2570734050i\)
\(L(1)\) \(\approx\) \(0.9712885611 + 0.2570734050i\)

Euler product

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

−19.79031680911092467727214826776, −18.9636804346120063025372558980, −18.75460664396575362033750172586, −17.893909434146575396023632898961, −17.38462805157262587344621039261, −16.20616304488921431667862745143, −15.66586900383988445632050071350, −14.76479130182366556809000184864, −13.97459590635589873648291643261, −12.97320922341207488122763255694, −12.68948541092199093390681871799, −11.50339379242813837212245866120, −11.3230016223794068128829141667, −9.98791772097824951366627183209, −9.38247503773231498270213978874, −8.657855520738706756008494459030, −7.98024962028823981169224255114, −7.5953322760567886010265626714, −6.31789891416922050965645187642, −5.69783822721138050002760360728, −4.159472947343353600250017801172, −3.339105921410447622301168857399, −2.6148841556874993832724629739, −1.83421820862125342074848014701, −1.01266563661393067630477135002, 0.68427885808981403006276542370, 1.9269825845384159422317058860, 2.64260700230892384288472679964, 3.91106664662566894364676097484, 4.75901230672183966003852421735, 5.313198593152429458687741599362, 6.7668170117180712853349384266, 7.33556407073799606255819252036, 7.88657746720150637732839382497, 8.78561491879747150701624371728, 9.551672413640176923902720295721, 10.02538065661498325726317460521, 10.82825029866428847271471685330, 11.4512505551181062492613717810, 12.84028297080299554206300599233, 13.69600784566471805481362944936, 14.33173304185094714607588980364, 14.89096303210809461746983623990, 15.720786922686519653928454567081, 16.27992400632600782717468698997, 16.929239079143305456887463917257, 17.93981906907022964439892879434, 18.25928884873347312905954086697, 19.39692821345376252587215689117, 19.9661911181136550837193843924

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