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Properties

Label 1-2015-2015.122-r1-0-0
Degree $1$
Conductor $2015$
Sign $-0.734 + 0.678i$
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  + (0.309 + 0.951i)2-s + (0.951 + 0.309i)3-s + (−0.809 + 0.587i)4-s + i·6-s + (0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (0.809 + 0.587i)9-s + (0.587 + 0.809i)11-s + (−0.951 + 0.309i)12-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (0.587 − 0.809i)17-s + (−0.309 + 0.951i)18-s + (−0.951 + 0.309i)19-s + (0.951 − 0.309i)21-s + (−0.587 + 0.809i)22-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (0.951 + 0.309i)3-s + (−0.809 + 0.587i)4-s + i·6-s + (0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (0.809 + 0.587i)9-s + (0.587 + 0.809i)11-s + (−0.951 + 0.309i)12-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (0.587 − 0.809i)17-s + (−0.309 + 0.951i)18-s + (−0.951 + 0.309i)19-s + (0.951 − 0.309i)21-s + (−0.587 + 0.809i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.734 + 0.678i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ -0.734 + 0.678i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.565847521 + 4.000657141i\)
\(L(\frac12)\) \(\approx\) \(1.565847521 + 4.000657141i\)
\(L(1)\) \(\approx\) \(1.421903732 + 1.172797473i\)
\(L(1)\) \(\approx\) \(1.421903732 + 1.172797473i\)

Euler product

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

−19.55041711088398378696779782820, −18.90694965373600144319902978019, −18.37286215367929288916959610798, −17.5951615128821903270081847878, −16.684446039702871606445802698004, −15.38409792643730306228899783082, −14.85270530690849515008046443356, −14.22583079365612690648476166261, −13.687557889883468262466387536161, −12.71601841081777453198882497002, −12.26054477709572275202634670969, −11.38646063953980120636186412776, −10.67176269671372096088938045195, −9.77066508638382099676414669599, −8.94422051510770592729515078442, −8.4148556394704509017836364028, −7.801052607050045749960914676026, −6.3217098279805310517559741632, −5.81234844107525317844094339456, −4.443653296999702927837124595481, −4.060483843502866674493353998330, −2.95776922647014626581130157764, −2.31068581294971904522305231612, −1.51091812472473997045129100606, −0.634228871230288993055078497001, 1.0311964279479936845989677487, 2.09786314400276626597501341529, 3.23674679094750644372661853949, 4.135139560782117319758128617747, 4.53822718859649189574986245314, 5.44926245827843078186477642329, 6.580628058382886470074854659456, 7.47556698284595707399033510792, 7.76841194889770734322232593302, 8.69649117135962133836246967019, 9.41897616620834047630464014339, 10.072121472336659863547346902201, 11.11073100600123918878844145635, 12.21730586794126159121250292380, 12.8775282821152848767586801652, 13.83833502279799873740636252628, 14.3160060843622395740418247694, 14.74851578636242280693272307224, 15.54300561022332690030865448169, 16.28207324095519017847801558908, 17.00026552813091721039707631175, 17.76126538564472127393289795410, 18.39551118069238047829956887335, 19.375061113173163956022454299781, 20.10962901261312580251927741021

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