↑

Properties

Label 2-1104-12.11-c1-0-17
Degree $2$
Conductor $1104$
Sign $0.630 - 0.776i$
Analytic cond. $8.81548$
Root an. cond. $2.96908$
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  + (0.618 + 1.61i)3-s − 3.18i·5-s + 3.68i·7-s + (−2.23 + 2.00i)9-s + 2.73·11-s + 5.46·13-s + (5.15 − 1.96i)15-s − 6.38i·17-s + 3.18i·19-s + (−5.96 + 2.27i)21-s − 23-s − 5.14·25-s + (−4.61 − 2.38i)27-s + 4.47i·29-s + 2.14i·31-s + ⋯
L(s)  = 1  + (0.356 + 0.934i)3-s − 1.42i·5-s + 1.39i·7-s + (−0.745 + 0.666i)9-s + 0.823·11-s + 1.51·13-s + (1.33 − 0.508i)15-s − 1.54i·17-s + 0.730i·19-s + (−1.30 + 0.497i)21-s − 0.208·23-s − 1.02·25-s + (−0.888 − 0.458i)27-s + 0.830i·29-s + 0.384i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1104\)    =    \(2^{4} \cdot 3 \cdot 23\)
Sign: $0.630 - 0.776i$
Analytic conductor: \(8.81548\)
Root analytic conductor: \(2.96908\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1104} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1104,\ (\ :1/2),\ 0.630 - 0.776i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.959288654\)
\(L(\frac12)\) \(\approx\) \(1.959288654\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.618 - 1.61i)T \)
23 \( 1 + T \)
good5 \( 1 + 3.18iT - 5T^{2} \)
7 \( 1 - 3.68iT - 7T^{2} \)
11 \( 1 - 2.73T + 11T^{2} \)
13 \( 1 - 5.46T + 13T^{2} \)
17 \( 1 + 6.38iT - 17T^{2} \)
19 \( 1 - 3.18iT - 19T^{2} \)
29 \( 1 - 4.47iT - 29T^{2} \)
31 \( 1 - 2.14iT - 31T^{2} \)
37 \( 1 - 6.87T + 37T^{2} \)
41 \( 1 - 8.70iT - 41T^{2} \)
43 \( 1 - 0.0893iT - 43T^{2} \)
47 \( 1 - 6.14T + 47T^{2} \)
53 \( 1 - 11.5iT - 53T^{2} \)
59 \( 1 + 5.70T + 59T^{2} \)
61 \( 1 - 4.05T + 61T^{2} \)
67 \( 1 + 11.1iT - 67T^{2} \)
71 \( 1 - 7.84T + 71T^{2} \)
73 \( 1 - 2.81T + 73T^{2} \)
79 \( 1 - 13.7iT - 79T^{2} \)
83 \( 1 - 5.66T + 83T^{2} \)
89 \( 1 + 14.8iT - 89T^{2} \)
97 \( 1 - 8.92T + 97T^{2} \)
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

−9.382759325766957542102382044920, −9.296875472067860836198521632347, −8.589266496972018385618123858180, −7.946058109406357214164927053713, −6.27585550633049629005592672939, −5.51966400725187255353826661358, −4.80983633890451666257154778945, −3.89579146798285944246404906557, −2.78446583853434316981091542935, −1.32982415491311485316827104360, 1.00255088050783440309269846925, 2.25406976284456313406976407718, 3.64969177146877795852765971946, 3.89904858904174100945582559547, 6.03957397109235101008015252203, 6.47606653842700246592756312599, 7.15387288216091648746712934870, 7.914093344695612901751787461193, 8.724794693810606241557788670164, 9.834021458844056170460462156074

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