↑

Properties

Label 2-1815-5.4-c1-0-29
Degree $2$
Conductor $1815$
Sign $-0.673 + 0.739i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.73i·2-s − i·3-s − 5.48·4-s + (−1.65 − 1.50i)5-s − 2.73·6-s + 4.20i·7-s + 9.52i·8-s − 9-s + (−4.11 + 4.52i)10-s + 5.48i·12-s + 0.0784i·13-s + 11.4·14-s + (−1.50 + 1.65i)15-s + 15.0·16-s − 0.0228i·17-s + 2.73i·18-s + ⋯
L(s)  = 1  − 1.93i·2-s − 0.577i·3-s − 2.74·4-s + (−0.739 − 0.673i)5-s − 1.11·6-s + 1.58i·7-s + 3.36i·8-s − 0.333·9-s + (−1.30 + 1.42i)10-s + 1.58i·12-s + 0.0217i·13-s + 3.07·14-s + (−0.388 + 0.426i)15-s + 3.77·16-s − 0.00554i·17-s + 0.644i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $-0.673 + 0.739i$
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: \((2,\ 1815,\ (\ :1/2),\ -0.673 + 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9814723091\)
\(L(\frac12)\) \(\approx\) \(0.9814723091\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.65 + 1.50i)T \)
11 \( 1 \)
good2 \( 1 + 2.73iT - 2T^{2} \)
7 \( 1 - 4.20iT - 7T^{2} \)
13 \( 1 - 0.0784iT - 13T^{2} \)
17 \( 1 + 0.0228iT - 17T^{2} \)
19 \( 1 - 1.26T + 19T^{2} \)
23 \( 1 + 6.12iT - 23T^{2} \)
29 \( 1 - 5.85T + 29T^{2} \)
31 \( 1 + 3.23T + 31T^{2} \)
37 \( 1 - 8.67iT - 37T^{2} \)
41 \( 1 + 3.84T + 41T^{2} \)
43 \( 1 - 0.859iT - 43T^{2} \)
47 \( 1 - 1.89iT - 47T^{2} \)
53 \( 1 + 4.55iT - 53T^{2} \)
59 \( 1 - 3.44T + 59T^{2} \)
61 \( 1 - 8.13T + 61T^{2} \)
67 \( 1 + 0.406iT - 67T^{2} \)
71 \( 1 - 8.40T + 71T^{2} \)
73 \( 1 + 15.1iT - 73T^{2} \)
79 \( 1 - 13.2T + 79T^{2} \)
83 \( 1 - 14.4iT - 83T^{2} \)
89 \( 1 - 8.19T + 89T^{2} \)
97 \( 1 - 1.64iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.025441784044249911300131628434, −8.468397959816151282740229672555, −8.018311709141369528874729774536, −6.42501962172179577699773374929, −5.22428574413816802640536859012, −4.79351859005214471973908830109, −3.60258124533712328872306257239, −2.74499120503993300317804079611, −1.94379823776531954730014515453, −0.76788218119759543165555564684, 0.59937940481318619709372700081, 3.49174525309868435318852028085, 3.93639812455478639661910738155, 4.74151207073028306668087523914, 5.62661539241010703819497516667, 6.64250748686968134260501297996, 7.21432636139907303904899232365, 7.70136777332195628352363351172, 8.434654051151425529444242376102, 9.380016563371798919000668173128

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