↑

Properties

Label 2-330-15.2-c1-0-19
Degree $2$
Conductor $330$
Sign $-0.737 + 0.675i$
Analytic cond. $2.63506$
Root an. cond. $1.62328$
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  + (0.707 − 0.707i)2-s + (0.751 − 1.56i)3-s − 1.00i·4-s + (−2.21 − 0.330i)5-s + (−0.571 − 1.63i)6-s + (−0.310 − 0.310i)7-s + (−0.707 − 0.707i)8-s + (−1.86 − 2.34i)9-s + (−1.79 + 1.32i)10-s − i·11-s + (−1.56 − 0.751i)12-s + (0.758 − 0.758i)13-s − 0.438·14-s + (−2.17 + 3.20i)15-s − 1.00·16-s + (1.97 − 1.97i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.434 − 0.900i)3-s − 0.500i·4-s + (−0.988 − 0.147i)5-s + (−0.233 − 0.667i)6-s + (−0.117 − 0.117i)7-s + (−0.250 − 0.250i)8-s + (−0.623 − 0.782i)9-s + (−0.568 + 0.420i)10-s − 0.301i·11-s + (−0.450 − 0.217i)12-s + (0.210 − 0.210i)13-s − 0.117·14-s + (−0.562 + 0.826i)15-s − 0.250·16-s + (0.478 − 0.478i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.737 + 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(330\)    =    \(2 \cdot 3 \cdot 5 \cdot 11\)
Sign: $-0.737 + 0.675i$
Analytic conductor: \(2.63506\)
Root analytic conductor: \(1.62328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{330} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 330,\ (\ :1/2),\ -0.737 + 0.675i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.555274 - 1.42866i\)
\(L(\frac12)\) \(\approx\) \(0.555274 - 1.42866i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-0.751 + 1.56i)T \)
5 \( 1 + (2.21 + 0.330i)T \)
11 \( 1 + iT \)
good7 \( 1 + (0.310 + 0.310i)T + 7iT^{2} \)
13 \( 1 + (-0.758 + 0.758i)T - 13iT^{2} \)
17 \( 1 + (-1.97 + 1.97i)T - 17iT^{2} \)
19 \( 1 + 1.37iT - 19T^{2} \)
23 \( 1 + (-2.73 - 2.73i)T + 23iT^{2} \)
29 \( 1 - 1.28T + 29T^{2} \)
31 \( 1 - 3.91T + 31T^{2} \)
37 \( 1 + (-5.63 - 5.63i)T + 37iT^{2} \)
41 \( 1 + 4.53iT - 41T^{2} \)
43 \( 1 + (6.86 - 6.86i)T - 43iT^{2} \)
47 \( 1 + (-7.19 + 7.19i)T - 47iT^{2} \)
53 \( 1 + (-4.23 - 4.23i)T + 53iT^{2} \)
59 \( 1 + 1.01T + 59T^{2} \)
61 \( 1 - 6.22T + 61T^{2} \)
67 \( 1 + (7.31 + 7.31i)T + 67iT^{2} \)
71 \( 1 + 3.60iT - 71T^{2} \)
73 \( 1 + (11.0 - 11.0i)T - 73iT^{2} \)
79 \( 1 + 12.3iT - 79T^{2} \)
83 \( 1 + (-9.76 - 9.76i)T + 83iT^{2} \)
89 \( 1 - 9.69T + 89T^{2} \)
97 \( 1 + (-2.59 - 2.59i)T + 97iT^{2} \)
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

−11.71853827566386830101932003027, −10.54422946567551811656197861476, −9.261828593223492397670682405925, −8.312203929813254172200543087851, −7.41910263569506954060141276571, −6.44522028068003269149194555735, −5.11693316512293279574525553747, −3.72751885185476863395903013548, −2.80375504359123440548150251944, −0.947387403783473197675048880488, 2.85447794357114358704714735937, 3.93112021196059697711300194142, 4.70252939776093494023759116804, 5.97081534123102032879152559508, 7.26093681055202836911246971326, 8.166804655515965917931269242435, 8.927346105977420914432180178116, 10.12103474396767721731683969392, 11.03859534821431149275269871840, 11.95165878546516639755170835883

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