↑

Properties

Label 2-133-133.12-c1-0-3
Degree $2$
Conductor $133$
Sign $-0.781 - 0.623i$
Analytic cond. $1.06201$
Root an. cond. $1.03053$
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  + 1.73i·2-s + (−0.5 + 0.866i)3-s − 0.999·4-s + (−1.49 − 0.866i)6-s + (−2 + 1.73i)7-s + 1.73i·8-s + (1 + 1.73i)9-s + (−1.5 − 2.59i)11-s + (0.499 − 0.866i)12-s + (2.5 − 4.33i)13-s + (−2.99 − 3.46i)14-s − 5·16-s + (1.5 + 0.866i)17-s + (−3 + 1.73i)18-s + (4 − 1.73i)19-s + ⋯
L(s)  = 1  + 1.22i·2-s + (−0.288 + 0.499i)3-s − 0.499·4-s + (−0.612 − 0.353i)6-s + (−0.755 + 0.654i)7-s + 0.612i·8-s + (0.333 + 0.577i)9-s + (−0.452 − 0.783i)11-s + (0.144 − 0.249i)12-s + (0.693 − 1.20i)13-s + (−0.801 − 0.925i)14-s − 1.25·16-s + (0.363 + 0.210i)17-s + (−0.707 + 0.408i)18-s + (0.917 − 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.781 - 0.623i$
Analytic conductor: \(1.06201\)
Root analytic conductor: \(1.03053\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 133,\ (\ :1/2),\ -0.781 - 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.328498 + 0.938976i\)
\(L(\frac12)\) \(\approx\) \(0.328498 + 0.938976i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2 - 1.73i)T \)
19 \( 1 + (-4 + 1.73i)T \)
good2 \( 1 - 1.73iT - 2T^{2} \)
3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 5T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 0.866i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-4.5 - 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 0.866i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.5 - 0.866i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.5 + 6.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.5 + 0.866i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + 10.3iT - 67T^{2} \)
71 \( 1 + (10.5 - 6.06i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.5 - 0.866i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 3.46iT - 79T^{2} \)
83 \( 1 - 17.3iT - 83T^{2} \)
89 \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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

−13.68470261771557086333796021964, −13.01846002779985000642267892509, −11.46107558547595672236187016168, −10.57333679883320162077858131777, −9.292859311003101820292074172861, −8.187147792334698397167219724853, −7.17622526206904361731278235053, −5.71899565014024450614597841021, −5.31440760399197553967929304769, −3.17504788315366696936069515708, 1.24442070893612661661689321331, 3.10285126953531171079661674310, 4.41864517465389456646413945425, 6.54144550217376443552867707505, 7.14925514281958595949169461262, 9.101287095415886157491507741689, 10.00547133047382846129366205943, 10.87730694452263436171332654717, 11.97958649163151395994016472701, 12.64534044218936075556824669031

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