↑

Properties

Label 2-1148-41.40-c1-0-6
Degree $2$
Conductor $1148$
Sign $0.236 - 0.971i$
Analytic cond. $9.16682$
Root an. cond. $3.02767$
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  + 1.03i·3-s + 2.44·5-s + i·7-s + 1.92·9-s + 5.06i·11-s − 2.54i·13-s + 2.53i·15-s + 5.30i·17-s − 1.14i·19-s − 1.03·21-s − 5.57·23-s + 0.980·25-s + 5.10i·27-s − 8.69i·29-s + 8.34·31-s + ⋯
L(s)  = 1  + 0.598i·3-s + 1.09·5-s + 0.377i·7-s + 0.641·9-s + 1.52i·11-s − 0.706i·13-s + 0.654i·15-s + 1.28i·17-s − 0.263i·19-s − 0.226·21-s − 1.16·23-s + 0.196·25-s + 0.982i·27-s − 1.61i·29-s + 1.49·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.236 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.236 - 0.971i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (1065, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.236 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.041558257\)
\(L(\frac12)\) \(\approx\) \(2.041558257\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - iT \)
41 \( 1 + (6.22 + 1.51i)T \)
good3 \( 1 - 1.03iT - 3T^{2} \)
5 \( 1 - 2.44T + 5T^{2} \)
11 \( 1 - 5.06iT - 11T^{2} \)
13 \( 1 + 2.54iT - 13T^{2} \)
17 \( 1 - 5.30iT - 17T^{2} \)
19 \( 1 + 1.14iT - 19T^{2} \)
23 \( 1 + 5.57T + 23T^{2} \)
29 \( 1 + 8.69iT - 29T^{2} \)
31 \( 1 - 8.34T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
43 \( 1 + 5.96T + 43T^{2} \)
47 \( 1 - 4.18iT - 47T^{2} \)
53 \( 1 - 6.93iT - 53T^{2} \)
59 \( 1 - 7.10T + 59T^{2} \)
61 \( 1 - 1.10T + 61T^{2} \)
67 \( 1 - 2.43iT - 67T^{2} \)
71 \( 1 + 6.81iT - 71T^{2} \)
73 \( 1 + 5.66T + 73T^{2} \)
79 \( 1 - 4.13iT - 79T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 - 2.55iT - 89T^{2} \)
97 \( 1 + 6.85iT - 97T^{2} \)
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

−10.08481828446526268008138361556, −9.541513435168822049182579588732, −8.380969113406536183663398096497, −7.57621807909568183763766266508, −6.40283840877648837268693981580, −5.81719595206811219141288400642, −4.71007016211480086268020275870, −4.06854915564118238881925113608, −2.53652084213567052678406128233, −1.65508093686480272992324535408, 0.950249308511799129717599226475, 2.04158344264765474592717744658, 3.21863486555721237280046503514, 4.47222705364282604627161005078, 5.54310489378634267865060972689, 6.38316001595563348718651208177, 6.94551557582497257814833300349, 8.004066934780759141001534866502, 8.777254669016198698985820653375, 9.827590561885530500167365147961

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