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Properties

Label 2-1470-35.27-c1-0-16
Degree $2$
Conductor $1470$
Sign $-0.391 - 0.920i$
Analytic cond. $11.7380$
Root an. cond. $3.42607$
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 + (−0.707 − 0.707i)3-s + 1.00i·4-s + (1.19 + 1.89i)5-s − 1.00i·6-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (−0.493 + 2.18i)10-s − 1.97·11-s + (0.707 − 0.707i)12-s + (2.19 + 2.19i)13-s + (0.493 − 2.18i)15-s − 1.00·16-s + (3.25 − 3.25i)17-s + (−0.707 + 0.707i)18-s + 4.21·19-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (0.533 + 0.845i)5-s − 0.408i·6-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (−0.156 + 0.689i)10-s − 0.596·11-s + (0.204 − 0.204i)12-s + (0.608 + 0.608i)13-s + (0.127 − 0.563i)15-s − 0.250·16-s + (0.789 − 0.789i)17-s + (−0.166 + 0.166i)18-s + 0.967·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.391 - 0.920i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ -0.391 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.875513678\)
\(L(\frac12)\) \(\approx\) \(1.875513678\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-1.19 - 1.89i)T \)
7 \( 1 \)
good11 \( 1 + 1.97T + 11T^{2} \)
13 \( 1 + (-2.19 - 2.19i)T + 13iT^{2} \)
17 \( 1 + (-3.25 + 3.25i)T - 17iT^{2} \)
19 \( 1 - 4.21T + 19T^{2} \)
23 \( 1 + (4.15 - 4.15i)T - 23iT^{2} \)
29 \( 1 - 8.94iT - 29T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + (1.96 + 1.96i)T + 37iT^{2} \)
41 \( 1 + 6.55iT - 41T^{2} \)
43 \( 1 + (6.33 - 6.33i)T - 43iT^{2} \)
47 \( 1 + (4.29 - 4.29i)T - 47iT^{2} \)
53 \( 1 + (-8.08 + 8.08i)T - 53iT^{2} \)
59 \( 1 + 4.20T + 59T^{2} \)
61 \( 1 - 11.1iT - 61T^{2} \)
67 \( 1 + (-3.89 - 3.89i)T + 67iT^{2} \)
71 \( 1 + 3.86T + 71T^{2} \)
73 \( 1 + (-10.7 - 10.7i)T + 73iT^{2} \)
79 \( 1 + 2.55iT - 79T^{2} \)
83 \( 1 + (9.52 + 9.52i)T + 83iT^{2} \)
89 \( 1 - 6.19T + 89T^{2} \)
97 \( 1 + (-1.48 + 1.48i)T - 97iT^{2} \)
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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

−9.825613797798311832360953813305, −8.900688036624079317278496005891, −7.78678925958653419573219101644, −7.18982656550069728246979431203, −6.54158692962582141697400264573, −5.57573499092984739537030919689, −5.17445193123951709119571239811, −3.69468685983248810856795226467, −2.87683226220512777148852070713, −1.57577883212803409005638569942, 0.67992936961794276013812327715, 1.96460143612802995366797665179, 3.25926032142893052102614462797, 4.21028260480503042830264407392, 5.10116851067258450995109079494, 5.74888079812468197142467404131, 6.35582332215216354564048929757, 7.88707170184464615160774821586, 8.456806365043460068350591134132, 9.623459461126766789118586633675

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