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Properties

Label 1-2015-2015.1143-r0-0-0
Degree $1$
Conductor $2015$
Sign $0.877 - 0.480i$
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  + (−0.587 − 0.809i)2-s + (−0.587 + 0.809i)3-s + (−0.309 + 0.951i)4-s + 6-s + (0.951 + 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.309 − 0.951i)9-s + (0.309 − 0.951i)11-s + (−0.587 − 0.809i)12-s + (−0.309 − 0.951i)14-s + (−0.809 − 0.587i)16-s + (−0.951 + 0.309i)17-s + (−0.587 + 0.809i)18-s + (−0.809 + 0.587i)19-s + (−0.809 + 0.587i)21-s + (−0.951 + 0.309i)22-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.587 + 0.809i)3-s + (−0.309 + 0.951i)4-s + 6-s + (0.951 + 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.309 − 0.951i)9-s + (0.309 − 0.951i)11-s + (−0.587 − 0.809i)12-s + (−0.309 − 0.951i)14-s + (−0.809 − 0.587i)16-s + (−0.951 + 0.309i)17-s + (−0.587 + 0.809i)18-s + (−0.809 + 0.587i)19-s + (−0.809 + 0.587i)21-s + (−0.951 + 0.309i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.877 - 0.480i$
Analytic conductor: \(9.35762\)
Root analytic conductor: \(9.35762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (1143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (0:\ ),\ 0.877 - 0.480i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8964284274 - 0.2293828582i\)
\(L(\frac12)\) \(\approx\) \(0.8964284274 - 0.2293828582i\)
\(L(1)\) \(\approx\) \(0.7075826035 - 0.08633343182i\)
\(L(1)\) \(\approx\) \(0.7075826035 - 0.08633343182i\)

Euler product

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

−19.8004901029799390591166432640, −19.06274667076290920238703832113, −18.26535876964207612647681624412, −17.7516265700986115953633290600, −17.08151273260378445270460584948, −16.84557775277698790302260767875, −15.48704583796179339855125218499, −15.10966724502378989034655366238, −14.16165059262068226813402930397, −13.492269316338872196123439519668, −12.77780921410714858806950283785, −11.6906430520796984591611698100, −11.04480749247005237487764547303, −10.474959718640848341725648295513, −9.31740022597930873190097165446, −8.66753235072637492412133001331, −7.68273258691610359704350061835, −7.26344299991823346979187343118, −6.55711818957067734543991491393, −5.73757846007718334551724186742, −4.74299803569967175003649949811, −4.411613540158080243793088617806, −2.39626023703546253361844728255, −1.68632828623953284661079467551, −0.77626251124095581434227468327, 0.6184667452675848558462576175, 1.713855964930996952669675148674, 2.69491490368604444486690569201, 3.77068122270265141510168449129, 4.30909516931664820942711188888, 5.2334134026522474813324294414, 6.08868110041371912735169997783, 7.14796110101994023346981959160, 8.24851564835518131733774139047, 8.885185567516768499552549980589, 9.32738800287831355026662811675, 10.67296267768376697464774771948, 10.80877190447447121796423645038, 11.45499226749534557658816913385, 12.27214130279403700879670916815, 12.9798751045756778526920713650, 14.07755191888682856186850208864, 14.804940232829012406924176918788, 15.636856618070912006811244421189, 16.57169187835813132043739777208, 16.99039540841385547150465009150, 17.73482804599665025674889050249, 18.35405013460990860452176956593, 19.13608450153150104292403995964, 19.94733780013982682741621941132

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