↑

Properties

Label 2-1380-345.344-c1-0-35
Degree $2$
Conductor $1380$
Sign $0.770 + 0.637i$
Analytic cond. $11.0193$
Root an. cond. $3.31954$
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  + (1.71 − 0.253i)3-s + (−0.923 − 2.03i)5-s + 2.63·7-s + (2.87 − 0.870i)9-s + 4.99·11-s + 4.22i·13-s + (−2.09 − 3.25i)15-s − 4.76i·17-s − 1.53i·19-s + (4.52 − 0.670i)21-s + (−4.57 + 1.44i)23-s + (−3.29 + 3.76i)25-s + (4.69 − 2.22i)27-s + 0.927i·29-s + 1.66·31-s + ⋯
L(s)  = 1  + (0.989 − 0.146i)3-s + (−0.412 − 0.910i)5-s + 0.997·7-s + (0.956 − 0.290i)9-s + 1.50·11-s + 1.17i·13-s + (−0.542 − 0.840i)15-s − 1.15i·17-s − 0.352i·19-s + (0.986 − 0.146i)21-s + (−0.953 + 0.301i)23-s + (−0.658 + 0.752i)25-s + (0.904 − 0.427i)27-s + 0.172i·29-s + 0.299·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.770 + 0.637i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ 0.770 + 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.718350972\)
\(L(\frac12)\) \(\approx\) \(2.718350972\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 + 0.253i)T \)
5 \( 1 + (0.923 + 2.03i)T \)
23 \( 1 + (4.57 - 1.44i)T \)
good7 \( 1 - 2.63T + 7T^{2} \)
11 \( 1 - 4.99T + 11T^{2} \)
13 \( 1 - 4.22iT - 13T^{2} \)
17 \( 1 + 4.76iT - 17T^{2} \)
19 \( 1 + 1.53iT - 19T^{2} \)
29 \( 1 - 0.927iT - 29T^{2} \)
31 \( 1 - 1.66T + 31T^{2} \)
37 \( 1 + 7.06T + 37T^{2} \)
41 \( 1 - 11.2iT - 41T^{2} \)
43 \( 1 - 0.983T + 43T^{2} \)
47 \( 1 - 8.19T + 47T^{2} \)
53 \( 1 + 10.8iT - 53T^{2} \)
59 \( 1 + 4.23iT - 59T^{2} \)
61 \( 1 - 12.0iT - 61T^{2} \)
67 \( 1 - 5.05T + 67T^{2} \)
71 \( 1 + 10.6iT - 71T^{2} \)
73 \( 1 + 11.4iT - 73T^{2} \)
79 \( 1 + 0.925iT - 79T^{2} \)
83 \( 1 + 12.6iT - 83T^{2} \)
89 \( 1 - 2.70T + 89T^{2} \)
97 \( 1 + 4.80T + 97T^{2} \)
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

−9.158233558079727380677501860362, −8.838953240516683133243175192906, −7.981440764962255784167193037299, −7.25011968694513908499676188499, −6.40903471590042858333145816688, −4.92828835612424619091609121060, −4.35941723759069669102780798257, −3.54624726785040660168697964427, −2.01519837528425565875111470385, −1.21574163621472348025301891220, 1.49072287323007749002288142494, 2.55018452032848941060995237244, 3.82757615529233990118202237463, 4.04574133085310107127394925733, 5.53458161323875476177026149671, 6.56975308169234914767031325924, 7.41140370104574001574447572554, 8.174303194407953211426594394262, 8.600339623347245001894308306482, 9.692084701463424565677334955332

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