↑

Properties

Label 2-2169-241.240-c1-0-51
Degree $2$
Conductor $2169$
Sign $0.817 + 0.575i$
Analytic cond. $17.3195$
Root an. cond. $4.16167$
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  + 1.18·2-s − 0.605·4-s − 3.46·5-s + 0.311i·7-s − 3.07·8-s − 4.08·10-s + 3.58i·11-s − 3.31i·13-s + 0.368i·14-s − 2.42·16-s + 5.50i·17-s + 0.614i·19-s + 2.09·20-s + 4.23i·22-s − 8.51i·23-s + ⋯
L(s)  = 1  + 0.835·2-s − 0.302·4-s − 1.54·5-s + 0.117i·7-s − 1.08·8-s − 1.29·10-s + 1.08i·11-s − 0.919i·13-s + 0.0983i·14-s − 0.605·16-s + 1.33i·17-s + 0.141i·19-s + 0.468·20-s + 0.903i·22-s − 1.77i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2169\)    =    \(3^{2} \cdot 241\)
Sign: $0.817 + 0.575i$
Analytic conductor: \(17.3195\)
Root analytic conductor: \(4.16167\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2169} (1927, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2169,\ (\ :1/2),\ 0.817 + 0.575i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.194762707\)
\(L(\frac12)\) \(\approx\) \(1.194762707\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
241 \( 1 + (-12.6 - 8.93i)T \)
good2 \( 1 - 1.18T + 2T^{2} \)
5 \( 1 + 3.46T + 5T^{2} \)
7 \( 1 - 0.311iT - 7T^{2} \)
11 \( 1 - 3.58iT - 11T^{2} \)
13 \( 1 + 3.31iT - 13T^{2} \)
17 \( 1 - 5.50iT - 17T^{2} \)
19 \( 1 - 0.614iT - 19T^{2} \)
23 \( 1 + 8.51iT - 23T^{2} \)
29 \( 1 - 0.779T + 29T^{2} \)
31 \( 1 + 6.09iT - 31T^{2} \)
37 \( 1 - 9.07iT - 37T^{2} \)
41 \( 1 - 4.52T + 41T^{2} \)
43 \( 1 + 11.1iT - 43T^{2} \)
47 \( 1 + 0.404T + 47T^{2} \)
53 \( 1 + 5.12T + 53T^{2} \)
59 \( 1 - 10.1T + 59T^{2} \)
61 \( 1 - 13.7T + 61T^{2} \)
67 \( 1 + 2.66T + 67T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 - 1.57iT - 73T^{2} \)
79 \( 1 - 4.04T + 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 + 0.994iT - 89T^{2} \)
97 \( 1 + 2.75T + 97T^{2} \)
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

−8.603164875705246245582907552688, −8.346934149381220991987951058333, −7.46811587481626492402807015751, −6.58292727604712178494898891629, −5.67843542805633641199757355307, −4.69144004408751995168931565235, −4.17009659833515682424920117474, −3.51402120724326059974570802961, −2.43309948665171437284336758805, −0.48756763783868864916574648581, 0.790693077931296002078761022767, 2.82422576589026652023737375846, 3.60636827125459729446139522312, 4.15158452593163356194121901170, 5.01477693442036284345916171951, 5.75748007705585656052428926077, 6.88381056325710927824340886968, 7.49523203354490016540106528110, 8.395137507403478125311663198588, 9.032945868161153202142976391233

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