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Properties

Label 2-1350-15.8-c1-0-10
Degree $2$
Conductor $1350$
Sign $0.793 + 0.608i$
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−1.22 + 1.22i)7-s + (0.707 − 0.707i)8-s + 1.73·14-s − 1.00·16-s + (−4.24 − 4.24i)17-s + i·19-s + (4.24 − 4.24i)23-s + (−1.22 − 1.22i)28-s + 10.3·29-s + 7·31-s + (0.707 + 0.707i)32-s + 6i·34-s + (−6.12 + 6.12i)37-s + (0.707 − 0.707i)38-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.462 + 0.462i)7-s + (0.250 − 0.250i)8-s + 0.462·14-s − 0.250·16-s + (−1.02 − 1.02i)17-s + 0.229i·19-s + (0.884 − 0.884i)23-s + (−0.231 − 0.231i)28-s + 1.92·29-s + 1.25·31-s + (0.125 + 0.125i)32-s + 1.02i·34-s + (−1.00 + 1.00i)37-s + (0.114 − 0.114i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.793 + 0.608i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.793 + 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.133948874\)
\(L(\frac12)\) \(\approx\) \(1.133948874\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 + (1.22 - 1.22i)T - 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (4.24 + 4.24i)T + 17iT^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 + (-4.24 + 4.24i)T - 23iT^{2} \)
29 \( 1 - 10.3T + 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 + (6.12 - 6.12i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-1.22 - 1.22i)T + 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (-8.48 + 8.48i)T - 53iT^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 5T + 61T^{2} \)
67 \( 1 + (-7.34 + 7.34i)T - 67iT^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (-8.57 - 8.57i)T + 73iT^{2} \)
79 \( 1 + 13iT - 79T^{2} \)
83 \( 1 + (-4.24 + 4.24i)T - 83iT^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + (6.12 - 6.12i)T - 97iT^{2} \)
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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

−9.619104318152617670605982358936, −8.648114603649068257224568841128, −8.316273312673296255551353033545, −6.90601455183879587416735049940, −6.57036717594970417962820623366, −5.18330017285717781859086250081, −4.36168948838174191312833113526, −3.04121627569113482079975580311, −2.39636281011616177037460258591, −0.77669000650196337738542581241, 0.904189285378749443931368489149, 2.38724638186623045499979595607, 3.69710165835026328341482135681, 4.67756900981516326734695398901, 5.69194282874015228610062445598, 6.68915294645861675725099355630, 7.04847019224525572853498248591, 8.233835330786692646308334861579, 8.737418144844792519859226030577, 9.653909014898541727073499932050

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