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Properties

Label 2-1350-45.38-c1-0-6
Degree $2$
Conductor $1350$
Sign $0.994 + 0.103i$
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (0.328 + 1.22i)7-s + (−0.707 + 0.707i)8-s + (−3 − 1.73i)11-s + (−0.328 + 1.22i)13-s + (−0.633 − 1.09i)14-s + (0.500 − 0.866i)16-s − 7.19i·19-s + (3.34 + 0.896i)22-s + (7.91 + 2.12i)23-s − 1.26i·26-s + (0.896 + 0.896i)28-s + (3.63 − 6.29i)29-s + (5.09 + 8.83i)31-s + (−0.258 + 0.965i)32-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.433 − 0.249i)4-s + (0.124 + 0.462i)7-s + (−0.249 + 0.249i)8-s + (−0.904 − 0.522i)11-s + (−0.0910 + 0.339i)13-s + (−0.169 − 0.293i)14-s + (0.125 − 0.216i)16-s − 1.65i·19-s + (0.713 + 0.191i)22-s + (1.65 + 0.442i)23-s − 0.248i·26-s + (0.169 + 0.169i)28-s + (0.674 − 1.16i)29-s + (0.915 + 1.58i)31-s + (−0.0457 + 0.170i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.994 + 0.103i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.994 + 0.103i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.124101501\)
\(L(\frac12)\) \(\approx\) \(1.124101501\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.328 - 1.22i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.328 - 1.22i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 + 7.19iT - 19T^{2} \)
23 \( 1 + (-7.91 - 2.12i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.63 + 6.29i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.09 - 8.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.55 - 1.55i)T - 37iT^{2} \)
41 \( 1 + (-1.5 + 0.866i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.24 - 1.67i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-5.79 + 1.55i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.55 + 1.55i)T - 53iT^{2} \)
59 \( 1 + (6.23 + 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.0 - 3.22i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 10.7iT - 71T^{2} \)
73 \( 1 + (-3.67 - 3.67i)T + 73iT^{2} \)
79 \( 1 + (-8.66 - 5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.76 + 6.57i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 8.66T + 89T^{2} \)
97 \( 1 + (3.79 + 14.1i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.470454402110483666645383876020, −8.708156115598524250664506257902, −8.203922982754568804753796211786, −7.11094842691423896287232256053, −6.56454888017615832674369762317, −5.36556900282792180817463440480, −4.79523656990469894205347827444, −3.15097453782845820646721446655, −2.35909832808079493635886552357, −0.77579919734473929263022979777, 0.954111221095402763950985667687, 2.29621793113475562922949223624, 3.32544831218366946679323630948, 4.49541008272329498118382526385, 5.46502045328475277076645030347, 6.50894076652585525221039230719, 7.45371330733027659851273653614, 7.941187986399720066499595427839, 8.819394711864168596232127973703, 9.702428926766309337168859770719

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