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Properties

Label 2-1350-15.8-c1-0-7
Degree $2$
Conductor $1350$
Sign $0.850 - 0.525i$
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 + (−1.44 + 1.44i)7-s + (0.707 − 0.707i)8-s − 1.09i·11-s + (−4.22 − 4.22i)13-s + 2.04·14-s − 1.00·16-s + (4.17 + 4.17i)17-s + 4.44i·19-s + (−0.775 + 0.775i)22-s + (4.48 − 4.48i)23-s + 5.97i·26-s + (−1.44 − 1.44i)28-s + 3.14·29-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.547 + 0.547i)7-s + (0.250 − 0.250i)8-s − 0.330i·11-s + (−1.17 − 1.17i)13-s + 0.547·14-s − 0.250·16-s + (1.01 + 1.01i)17-s + 1.02i·19-s + (−0.165 + 0.165i)22-s + (0.936 − 0.936i)23-s + 1.17i·26-s + (−0.273 − 0.273i)28-s + 0.584·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.850 - 0.525i$
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: \((2,\ 1350,\ (\ :1/2),\ 0.850 - 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9816381952\)
\(L(\frac12)\) \(\approx\) \(0.9816381952\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.44 - 1.44i)T - 7iT^{2} \)
11 \( 1 + 1.09iT - 11T^{2} \)
13 \( 1 + (4.22 + 4.22i)T + 13iT^{2} \)
17 \( 1 + (-4.17 - 4.17i)T + 17iT^{2} \)
19 \( 1 - 4.44iT - 19T^{2} \)
23 \( 1 + (-4.48 + 4.48i)T - 23iT^{2} \)
29 \( 1 - 3.14T + 29T^{2} \)
31 \( 1 + 1.44T + 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 - 4.87iT - 41T^{2} \)
43 \( 1 + (-7.22 - 7.22i)T + 43iT^{2} \)
47 \( 1 + (-7.31 - 7.31i)T + 47iT^{2} \)
53 \( 1 + (5.65 - 5.65i)T - 53iT^{2} \)
59 \( 1 + 2.82T + 59T^{2} \)
61 \( 1 + 8.44T + 61T^{2} \)
67 \( 1 + (2 - 2i)T - 67iT^{2} \)
71 \( 1 - 13.9iT - 71T^{2} \)
73 \( 1 + (-4.44 - 4.44i)T + 73iT^{2} \)
79 \( 1 - 5.44iT - 79T^{2} \)
83 \( 1 + (-10.2 + 10.2i)T - 83iT^{2} \)
89 \( 1 - 17.4T + 89T^{2} \)
97 \( 1 + (8.44 - 8.44i)T - 97iT^{2} \)
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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

−9.743584453556233508353433652679, −8.998444112748535737630952265102, −8.039876861911468151640327741824, −7.59773164270785052987779680286, −6.30023162150392864711127522162, −5.62801603461650053274762145209, −4.47721054556024230477444769867, −3.21417846065559138306633367289, −2.63466959303357571749364922695, −1.08264834605008568852995560002, 0.56959322387944410185735935517, 2.15384539760650816871920239587, 3.39123738519958149344029146984, 4.66221612915879196103838012391, 5.30473274194676816442431372672, 6.56255281204160970744002967954, 7.24259506660472686014800444190, 7.55740647140918706378847322305, 9.051815829365069031504201014932, 9.333887327933462977308143403775

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