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Properties

Label 2-1470-35.13-c1-0-7
Degree $2$
Conductor $1470$
Sign $0.414 - 0.910i$
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 + (−2.14 + 0.625i)5-s + 1.00i·6-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (−1.07 + 1.96i)10-s + 2.07·11-s + (0.707 + 0.707i)12-s + (0.326 − 0.326i)13-s + (1.07 − 1.96i)15-s − 1.00·16-s + (1.26 + 1.26i)17-s + (−0.707 − 0.707i)18-s − 4.37·19-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s − 0.500i·4-s + (−0.960 + 0.279i)5-s + 0.408i·6-s + (−0.250 − 0.250i)8-s − 0.333i·9-s + (−0.340 + 0.619i)10-s + 0.625·11-s + (0.204 + 0.204i)12-s + (0.0906 − 0.0906i)13-s + (0.277 − 0.506i)15-s − 0.250·16-s + (0.307 + 0.307i)17-s + (−0.166 − 0.166i)18-s − 1.00·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.414 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.414 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.414 - 0.910i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (1273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.414 - 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.147603728\)
\(L(\frac12)\) \(\approx\) \(1.147603728\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.14 - 0.625i)T \)
7 \( 1 \)
good11 \( 1 - 2.07T + 11T^{2} \)
13 \( 1 + (-0.326 + 0.326i)T - 13iT^{2} \)
17 \( 1 + (-1.26 - 1.26i)T + 17iT^{2} \)
19 \( 1 + 4.37T + 19T^{2} \)
23 \( 1 + (-0.635 - 0.635i)T + 23iT^{2} \)
29 \( 1 - 0.0288iT - 29T^{2} \)
31 \( 1 - 8.03iT - 31T^{2} \)
37 \( 1 + (8.07 - 8.07i)T - 37iT^{2} \)
41 \( 1 - 10.6iT - 41T^{2} \)
43 \( 1 + (-2.50 - 2.50i)T + 43iT^{2} \)
47 \( 1 + (-0.525 - 0.525i)T + 47iT^{2} \)
53 \( 1 + (-7.22 - 7.22i)T + 53iT^{2} \)
59 \( 1 - 13.4T + 59T^{2} \)
61 \( 1 + 9.84iT - 61T^{2} \)
67 \( 1 + (3.33 - 3.33i)T - 67iT^{2} \)
71 \( 1 - 2.27T + 71T^{2} \)
73 \( 1 + (8.14 - 8.14i)T - 73iT^{2} \)
79 \( 1 - 8.01iT - 79T^{2} \)
83 \( 1 + (4.26 - 4.26i)T - 83iT^{2} \)
89 \( 1 - 0.0197T + 89T^{2} \)
97 \( 1 + (-5.65 - 5.65i)T + 97iT^{2} \)
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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

−9.952502637243478510383002230199, −8.845510221377485990927966809789, −8.200687287665449134435641045324, −6.96219453857515659825927516328, −6.41243023471875377877994928349, −5.29689545441374361049284373594, −4.43705470277517000114870215090, −3.74261433499505723233938891008, −2.88323589055990096311466968525, −1.26142688709279443239138064733, 0.45037466615111791469840454794, 2.16494474825439551165328465248, 3.65982576637039416137305393052, 4.24267448919760117849387320435, 5.27726314753646635315238530313, 6.05820993173386799674581590945, 7.08034142315261209693455835769, 7.42883892531575982597382648404, 8.526355796566296672706995594725, 8.968694035649611216994891861771

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