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Properties

Label 2-1350-15.8-c1-0-18
Degree $2$
Conductor $1350$
Sign $-0.437 + 0.899i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.437 + 0.899i$
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: \((2,\ 1350,\ (\ :1/2),\ -0.437 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.044265200\)
\(L(\frac12)\) \(\approx\) \(1.044265200\)
\(L(\frac{3}{2})\) 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

−9.265976322787672410733787142119, −8.764304954639611809708428761377, −7.70581614263989836076679329103, −7.17324719902625095893494538193, −6.14045086861577677841951339377, −4.88956214321609264556287449138, −4.17806872238576729423685088755, −2.97404724489201598206639185458, −1.94094925174734476547812087214, −0.53038890879752740608558706836, 1.38822748557989546825409967720, 2.55043497088047339196635734269, 3.98166576338394932210969768525, 4.98142418280382511875373978787, 5.82867178915113534617917488329, 6.63014424433102306854535550566, 7.52826539603289712069000422792, 8.288518530224666217556067836335, 8.981745418593224473826828974507, 9.642730809841484461149969709354

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