Properties

Label 2-1368-19.11-c1-0-3
Degree $2$
Conductor $1368$
Sign $0.0547 - 0.998i$
Analytic cond. $10.9235$
Root an. cond. $3.30507$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.347 − 0.601i)5-s + 0.305·7-s − 4.82·11-s + (−0.5 − 0.866i)13-s + (−3.75 + 6.51i)17-s + (3.06 + 3.10i)19-s + (0.347 + 0.601i)23-s + (2.25 + 3.91i)25-s + (5.06 + 8.77i)29-s − 1.82·31-s + (0.106 − 0.183i)35-s + 6.51·37-s + (2.69 − 4.66i)41-s + (−1.84 + 3.19i)43-s + (−3 − 5.19i)47-s + ⋯
L(s)  = 1  + (0.155 − 0.269i)5-s + 0.115·7-s − 1.45·11-s + (−0.138 − 0.240i)13-s + (−0.911 + 1.57i)17-s + (0.702 + 0.711i)19-s + (0.0724 + 0.125i)23-s + (0.451 + 0.782i)25-s + (0.940 + 1.62i)29-s − 0.327·31-s + (0.0179 − 0.0310i)35-s + 1.07·37-s + (0.420 − 0.728i)41-s + (−0.281 + 0.487i)43-s + (−0.437 − 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0547 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0547 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.0547 - 0.998i$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ 0.0547 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.143204636\)
\(L(\frac12)\) \(\approx\) \(1.143204636\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + (-3.06 - 3.10i)T \)
good5 \( 1 + (-0.347 + 0.601i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 0.305T + 7T^{2} \)
11 \( 1 + 4.82T + 11T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.75 - 6.51i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.347 - 0.601i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.06 - 8.77i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.82T + 31T^{2} \)
37 \( 1 - 6.51T + 37T^{2} \)
41 \( 1 + (-2.69 + 4.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.84 - 3.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.71 - 4.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.04 - 3.53i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.194 + 0.337i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.91 - 6.77i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.45 + 9.44i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.19 + 3.80i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.21 - 12.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.739T + 83T^{2} \)
89 \( 1 + (0.411 + 0.712i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.45 - 9.44i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.861890324982280275189190468406, −8.874576128654966283289485470097, −8.191855120054440758240599646379, −7.48858163325736010576962980179, −6.46628809403850662090643797355, −5.49862487186304811298833459574, −4.90078291673612469805789538801, −3.71096428534882567228945134019, −2.65091461773479678851816683072, −1.42564037333114073648210165034, 0.46707141351337917191615402969, 2.40821147042629645041435822305, 2.89965496176559572892509945095, 4.53342190598774434080154771440, 5.01941661437313986176489374019, 6.14881390644942302402376883029, 6.99511085983117686360823090769, 7.75846745321920192609229206641, 8.524598924143990088850319151014, 9.597740319249801648853037428736

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