Properties

Label 2-1815-5.4-c1-0-98
Degree $2$
Conductor $1815$
Sign $-0.891 - 0.453i$
Analytic cond. $14.4928$
Root an. cond. $3.80694$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.27i·2-s i·3-s + 0.384·4-s + (1.01 − 1.99i)5-s − 1.27·6-s − 0.483i·7-s − 3.03i·8-s − 9-s + (−2.53 − 1.28i)10-s − 0.384i·12-s + 3.14i·13-s − 0.614·14-s + (−1.99 − 1.01i)15-s − 3.08·16-s + 2.51i·17-s + 1.27i·18-s + ⋯
L(s)  = 1  − 0.898i·2-s − 0.577i·3-s + 0.192·4-s + (0.453 − 0.891i)5-s − 0.518·6-s − 0.182i·7-s − 1.07i·8-s − 0.333·9-s + (−0.801 − 0.407i)10-s − 0.111i·12-s + 0.872i·13-s − 0.164·14-s + (−0.514 − 0.261i)15-s − 0.770·16-s + 0.608i·17-s + 0.299i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $-0.891 - 0.453i$
Analytic conductor: \(14.4928\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1815,\ (\ :1/2),\ -0.891 - 0.453i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.632718751\)
\(L(\frac12)\) \(\approx\) \(1.632718751\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 + (-1.01 + 1.99i)T \)
11 \( 1 \)
good2 \( 1 + 1.27iT - 2T^{2} \)
7 \( 1 + 0.483iT - 7T^{2} \)
13 \( 1 - 3.14iT - 13T^{2} \)
17 \( 1 - 2.51iT - 17T^{2} \)
19 \( 1 + 3.72T + 19T^{2} \)
23 \( 1 + 8.53iT - 23T^{2} \)
29 \( 1 + 9.98T + 29T^{2} \)
31 \( 1 + 5.78T + 31T^{2} \)
37 \( 1 - 3.20iT - 37T^{2} \)
41 \( 1 - 7.86T + 41T^{2} \)
43 \( 1 + 5.97iT - 43T^{2} \)
47 \( 1 + 6.13iT - 47T^{2} \)
53 \( 1 + 1.58iT - 53T^{2} \)
59 \( 1 - 0.377T + 59T^{2} \)
61 \( 1 + 3.00T + 61T^{2} \)
67 \( 1 + 5.98iT - 67T^{2} \)
71 \( 1 - 8.34T + 71T^{2} \)
73 \( 1 + 0.151iT - 73T^{2} \)
79 \( 1 - 7.48T + 79T^{2} \)
83 \( 1 - 4.05iT - 83T^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 + 3.60iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.982940567398907871689888943313, −8.174954481509282601218897069874, −7.13798078099422930065217339064, −6.44814419267189688939463917323, −5.65780543293106493020973468229, −4.43355326740789072569710113500, −3.72474834572379537010009976787, −2.23011269518760088575006300330, −1.84663638806089236219387045953, −0.54384407581925886299424281002, 1.99421733611488499017889328086, 2.93089938672629379144740292191, 3.88508803940962631563180112478, 5.32616921178396085297730606877, 5.67821451100175414250341985790, 6.45598179963736064754717636370, 7.58451401368634449922187797820, 7.64504858936953761856101783863, 9.070104491481453437883796187458, 9.490327921272002464900656103164

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