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Properties

Label 2-1350-9.7-c1-0-9
Degree $2$
Conductor $1350$
Sign $0.766 + 0.642i$
Analytic cond. $10.7798$
Root an. cond. $3.28326$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)7-s − 0.999·8-s + (−1 + 1.73i)11-s + (3 + 5.19i)13-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s + 2·17-s + 6·19-s + (0.999 + 1.73i)22-s + (0.5 + 0.866i)23-s + 6·26-s − 0.999·28-s + (4.5 − 7.79i)29-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.188 − 0.327i)7-s − 0.353·8-s + (−0.301 + 0.522i)11-s + (0.832 + 1.44i)13-s + (−0.133 − 0.231i)14-s + (−0.125 + 0.216i)16-s + 0.485·17-s + 1.37·19-s + (0.213 + 0.369i)22-s + (0.104 + 0.180i)23-s + 1.17·26-s − 0.188·28-s + (0.835 − 1.44i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.102054966\)
\(L(\frac12)\) \(\approx\) \(2.102054966\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3 - 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (5.5 + 9.52i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.5 - 6.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + (-6 + 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.5 - 9.52i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + T + 89T^{2} \)
97 \( 1 + (-4 + 6.92i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.661551721970156061428062779576, −8.904348762536853763362943191910, −7.899356743076805658758081521360, −7.03641321811563206808671022494, −6.12705299990550386674174035839, −5.13509841862378702042054862255, −4.28798078596763913223665098551, −3.46938548816308408760051880023, −2.23833209356571198125667153809, −1.11919988156753806813512814014, 1.03038065239971203891007142409, 2.93027409699647923000437327128, 3.49539953319052853498205150878, 4.94421653299149623475874815360, 5.49168491834556954928837097563, 6.26795048789240915354021246679, 7.28249958751412230256077157485, 8.153927211192121552050246523868, 8.534239114370644041235166018900, 9.652709068434193011728793665670

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