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Properties

Label 2-2205-35.19-c0-0-1
Degree $2$
Conductor $2205$
Sign $0.611 + 0.791i$
Analytic cond. $1.10043$
Root an. cond. $1.04901$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.60 − 0.923i)2-s + (1.20 + 2.09i)4-s + (0.5 − 0.866i)5-s − 2.61i·8-s + (−1.60 + 0.923i)10-s + (−1.20 + 2.09i)16-s + (0.707 + 1.22i)17-s + (1.60 + 0.923i)19-s + 2.41·20-s + (0.662 + 0.382i)23-s + (−0.499 − 0.866i)25-s + (−0.662 + 0.382i)31-s + (1.60 − 0.923i)32-s − 2.61i·34-s + (−1.70 − 2.95i)38-s + ⋯
L(s)  = 1  + (−1.60 − 0.923i)2-s + (1.20 + 2.09i)4-s + (0.5 − 0.866i)5-s − 2.61i·8-s + (−1.60 + 0.923i)10-s + (−1.20 + 2.09i)16-s + (0.707 + 1.22i)17-s + (1.60 + 0.923i)19-s + 2.41·20-s + (0.662 + 0.382i)23-s + (−0.499 − 0.866i)25-s + (−0.662 + 0.382i)31-s + (1.60 − 0.923i)32-s − 2.61i·34-s + (−1.70 − 2.95i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.611 + 0.791i$
Analytic conductor: \(1.10043\)
Root analytic conductor: \(1.04901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :0),\ 0.611 + 0.791i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6383962895\)
\(L(\frac12)\) \(\approx\) \(0.6383962895\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (1.60 + 0.923i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.60 - 0.923i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.662 + 0.382i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.224887734457391247749811513803, −8.559323050358414445438397248796, −7.903370997108858904450106023481, −7.27415704698650791672999080727, −6.07106717399957774761630868236, −5.21617768139091128397103862390, −3.84854194068896696135350464834, −3.04267371568733881565182325294, −1.75260935621815437566576555892, −1.15380105608693531645783570027, 0.957181908591783691697556661902, 2.29451056143379814375245507039, 3.24425301925562625881321084046, 5.11495584050188687907701113841, 5.62939128489970083663993400742, 6.65487573701729181986409905768, 7.18081123947480078110893183065, 7.62024961627470552871477201689, 8.648327095954035604848539278061, 9.418145418113107795518938409519

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