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Properties

Label 1-2015-2015.1877-r1-0-0
Degree $1$
Conductor $2015$
Sign $-0.611 + 0.791i$
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  + (−0.809 − 0.587i)2-s + (0.994 − 0.104i)3-s + (0.309 + 0.951i)4-s + (−0.866 − 0.5i)6-s + (0.978 + 0.207i)7-s + (0.309 − 0.951i)8-s + (0.978 − 0.207i)9-s + (−0.743 − 0.669i)11-s + (0.406 + 0.913i)12-s + (−0.669 − 0.743i)14-s + (−0.809 + 0.587i)16-s + (−0.743 + 0.669i)17-s + (−0.913 − 0.406i)18-s + (0.406 + 0.913i)19-s + (0.994 + 0.104i)21-s + (0.207 + 0.978i)22-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.994 − 0.104i)3-s + (0.309 + 0.951i)4-s + (−0.866 − 0.5i)6-s + (0.978 + 0.207i)7-s + (0.309 − 0.951i)8-s + (0.978 − 0.207i)9-s + (−0.743 − 0.669i)11-s + (0.406 + 0.913i)12-s + (−0.669 − 0.743i)14-s + (−0.809 + 0.587i)16-s + (−0.743 + 0.669i)17-s + (−0.913 − 0.406i)18-s + (0.406 + 0.913i)19-s + (0.994 + 0.104i)21-s + (0.207 + 0.978i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.611 + 0.791i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (1877, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ -0.611 + 0.791i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1639358784 + 0.3339510163i\)
\(L(\frac12)\) \(\approx\) \(0.1639358784 + 0.3339510163i\)
\(L(1)\) \(\approx\) \(0.9251900949 - 0.1901085498i\)
\(L(1)\) \(\approx\) \(0.9251900949 - 0.1901085498i\)

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 \( 1 + (0.994 - 0.104i)T \)
7 \( 1 + (0.978 + 0.207i)T \)
11 \( 1 + (-0.743 - 0.669i)T \)
17 \( 1 + (-0.743 + 0.669i)T \)
19 \( 1 + (0.406 + 0.913i)T \)
23 \( 1 + (-0.951 - 0.309i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.994 + 0.104i)T \)
43 \( 1 + (-0.406 - 0.913i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (-0.207 - 0.978i)T \)
59 \( 1 + (-0.994 + 0.104i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.207 + 0.978i)T \)
73 \( 1 + (-0.669 + 0.743i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (-0.104 + 0.994i)T \)
89 \( 1 + (-0.951 + 0.309i)T \)
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

−19.587595542312281547494389621461, −18.5876216426307818560000100373, −17.963190543490917923476674467608, −17.642354416808591075811907282215, −16.46660322747050277738991603919, −15.69615645195828982848061565670, −15.29273249325173457362800084806, −14.46854631933848875784609724231, −13.89974235765842494713644102054, −13.18341517997802163828675548397, −12.027514590767321035599582650000, −10.9597992047903240766373057266, −10.529691111984648822536298389154, −9.420804095025242964463373838196, −9.11225335026876414455598143504, −8.1415630015064661109510453565, −7.45568921594977869336881570376, −7.19150201564800692785094208456, −5.86240408328465629016724002649, −4.847129279100435506493029330135, −4.368249827213639503413891624652, −2.88309306172097424822813540705, −2.09508177269163223706170723974, −1.37370969491692965341172058305, −0.06653603078585614289819106459, 1.24133215943785739505430140056, 1.99228151290629552610969921392, 2.632316012583008595360606715947, 3.67096562082444331146802514697, 4.284691810583939311289455485897, 5.54723986280575154749517117177, 6.668871094594357044226971717437, 7.75723296197564919161498571885, 8.0848786668310306175197547378, 8.678793756468962608781430967642, 9.50759393188129853965199488125, 10.36807467921189034991061259321, 10.919327483686732354911343956313, 11.87896226629381585999481359492, 12.53988280957317708482163309004, 13.4665338763016891777969532756, 13.97461355977136059539884387163, 15.02141223640456546689570012932, 15.6078098522469347952594156055, 16.43354293856066777171957173029, 17.312478201985977408011022137923, 18.1515733000330190090685689604, 18.600526764951526114665576196711, 19.175349412510137675519791173628, 20.19022296109575547105236556336

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