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Properties

Label 1-2015-2015.348-r1-0-0
Degree $1$
Conductor $2015$
Sign $0.997 + 0.0640i$
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  + (0.994 − 0.104i)2-s + (−0.994 − 0.104i)3-s + (0.978 − 0.207i)4-s − 6-s + (0.951 + 0.309i)7-s + (0.951 − 0.309i)8-s + (0.978 + 0.207i)9-s + (−0.309 + 0.951i)11-s + (−0.994 + 0.104i)12-s + (0.978 + 0.207i)14-s + (0.913 − 0.406i)16-s + (0.951 − 0.309i)17-s + (0.994 + 0.104i)18-s + (−0.809 + 0.587i)19-s + (−0.913 − 0.406i)21-s + (−0.207 + 0.978i)22-s + ⋯
L(s)  = 1  + (0.994 − 0.104i)2-s + (−0.994 − 0.104i)3-s + (0.978 − 0.207i)4-s − 6-s + (0.951 + 0.309i)7-s + (0.951 − 0.309i)8-s + (0.978 + 0.207i)9-s + (−0.309 + 0.951i)11-s + (−0.994 + 0.104i)12-s + (0.978 + 0.207i)14-s + (0.913 − 0.406i)16-s + (0.951 − 0.309i)17-s + (0.994 + 0.104i)18-s + (−0.809 + 0.587i)19-s + (−0.913 − 0.406i)21-s + (−0.207 + 0.978i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.997 + 0.0640i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (348, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ 0.997 + 0.0640i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.298050930 + 0.1378896768i\)
\(L(\frac12)\) \(\approx\) \(4.298050930 + 0.1378896768i\)
\(L(1)\) \(\approx\) \(1.826024847 - 0.04285504201i\)
\(L(1)\) \(\approx\) \(1.826024847 - 0.04285504201i\)

Euler product

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

−19.8165593772161499910715883669, −19.08590961326084988041155761209, −18.1439224937485853520926219387, −17.3169206245662938028419028750, −16.76462284273396640573253594568, −16.16262156166624390373896937446, −15.20934350673779975944009676023, −14.78761201723649170065198931901, −13.65850228199329433062768071099, −13.28723760762848264076318144588, −12.277364972649402172621202145870, −11.64949340595700191563715915537, −10.984363013780856244547723796852, −10.59202304686849513543408116045, −9.48888306793473009334970149535, −8.04525184793216359049035131375, −7.67643704888880846151471703902, −6.558523522418811730681726282051, −5.92195101877595032111552072395, −5.24059966412027015530327794824, −4.51839932178978932283132086236, −3.80868905105794044528026592958, −2.74678576863291798863730554742, −1.56501152313030894684622976943, −0.76374803294509481850521359896, 0.823436209026845101675173217757, 1.78837162505318488806923931109, 2.47187792499824682847940400324, 3.859860805411684699726979658929, 4.58101294443611015948518478550, 5.231314315736640343521894932091, 5.80728805905103933984192078897, 6.78553863242790109652414242517, 7.422444097562404418975037802798, 8.23750426114942773792245070782, 9.60900074613388226329847734267, 10.61916861603373509408017056774, 10.82083643725943864540261561543, 11.95582739534060014495709085753, 12.30344627394919920643412857671, 12.86727208635606547853288669571, 13.945282517492347969094106974, 14.69672788126421257106897962794, 15.20887410120524820916675507535, 16.10602916339629657681427314288, 16.790226078334366752230978500356, 17.43945512184007322390716459209, 18.43696789407331647688492018112, 18.79318268957451248811286057594, 20.08762154748112242345707935066

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