Properties

Label 2-1110-1.1-c1-0-14
Degree $2$
Conductor $1110$
Sign $1$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s + 0.487·7-s + 8-s + 9-s + 10-s − 5.20·11-s + 12-s + 4.12·13-s + 0.487·14-s + 15-s + 16-s + 3.67·17-s + 18-s + 0.430·19-s + 20-s + 0.487·21-s − 5.20·22-s + 7.31·23-s + 24-s + 25-s + 4.12·26-s + 27-s + 0.487·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.184·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.57·11-s + 0.288·12-s + 1.14·13-s + 0.130·14-s + 0.258·15-s + 0.250·16-s + 0.891·17-s + 0.235·18-s + 0.0986·19-s + 0.223·20-s + 0.106·21-s − 1.11·22-s + 1.52·23-s + 0.204·24-s + 0.200·25-s + 0.808·26-s + 0.192·27-s + 0.0921·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $1$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.385529281\)
\(L(\frac12)\) \(\approx\) \(3.385529281\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
37 \( 1 + T \)
good7 \( 1 - 0.487T + 7T^{2} \)
11 \( 1 + 5.20T + 11T^{2} \)
13 \( 1 - 4.12T + 13T^{2} \)
17 \( 1 - 3.67T + 17T^{2} \)
19 \( 1 - 0.430T + 19T^{2} \)
23 \( 1 - 7.31T + 23T^{2} \)
29 \( 1 + 6.77T + 29T^{2} \)
31 \( 1 - 7.80T + 31T^{2} \)
41 \( 1 + 9.80T + 41T^{2} \)
43 \( 1 - 4.77T + 43T^{2} \)
47 \( 1 - 3.52T + 47T^{2} \)
53 \( 1 + 7.67T + 53T^{2} \)
59 \( 1 + 0.974T + 59T^{2} \)
61 \( 1 + 14.5T + 61T^{2} \)
67 \( 1 + 1.19T + 67T^{2} \)
71 \( 1 - 8.58T + 71T^{2} \)
73 \( 1 + 7.15T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 16.5T + 83T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 + 7.16T + 97T^{2} \)
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

−9.965134280770507071939025302068, −8.959573761759463273629336769266, −8.083090202187354978017654106298, −7.42965784754398170579912385024, −6.34431787794355415468557712204, −5.44978487914199118205939634699, −4.74750980364201892268137178643, −3.42752121929457007523382079861, −2.75847261528225599328467992197, −1.46119511378691107243432288840, 1.46119511378691107243432288840, 2.75847261528225599328467992197, 3.42752121929457007523382079861, 4.74750980364201892268137178643, 5.44978487914199118205939634699, 6.34431787794355415468557712204, 7.42965784754398170579912385024, 8.083090202187354978017654106298, 8.959573761759463273629336769266, 9.965134280770507071939025302068

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