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Properties

Label 2-1470-35.27-c1-0-4
Degree $2$
Conductor $1470$
Sign $-0.809 - 0.586i$
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 + (1.99 + 1.00i)5-s − 1.00i·6-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (−0.700 − 2.12i)10-s − 3.10·11-s + (−0.707 + 0.707i)12-s + (−3.40 − 3.40i)13-s + (0.700 + 2.12i)15-s − 1.00·16-s + (−3.76 + 3.76i)17-s + (0.707 − 0.707i)18-s − 7.23·19-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.408 + 0.408i)3-s + 0.500i·4-s + (0.892 + 0.450i)5-s − 0.408i·6-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (−0.221 − 0.671i)10-s − 0.937·11-s + (−0.204 + 0.204i)12-s + (−0.945 − 0.945i)13-s + (0.180 + 0.548i)15-s − 0.250·16-s + (−0.913 + 0.913i)17-s + (0.166 − 0.166i)18-s − 1.65·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.809 - 0.586i$
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: \((2,\ 1470,\ (\ :1/2),\ -0.809 - 0.586i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5416963519\)
\(L(\frac12)\) \(\approx\) \(0.5416963519\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.99 - 1.00i)T \)
7 \( 1 \)
good11 \( 1 + 3.10T + 11T^{2} \)
13 \( 1 + (3.40 + 3.40i)T + 13iT^{2} \)
17 \( 1 + (3.76 - 3.76i)T - 17iT^{2} \)
19 \( 1 + 7.23T + 19T^{2} \)
23 \( 1 + (3.72 - 3.72i)T - 23iT^{2} \)
29 \( 1 + 4.49iT - 29T^{2} \)
31 \( 1 - 9.22iT - 31T^{2} \)
37 \( 1 + (2.54 + 2.54i)T + 37iT^{2} \)
41 \( 1 + 2.51iT - 41T^{2} \)
43 \( 1 + (3.86 - 3.86i)T - 43iT^{2} \)
47 \( 1 + (-2.88 + 2.88i)T - 47iT^{2} \)
53 \( 1 + (-2.43 + 2.43i)T - 53iT^{2} \)
59 \( 1 - 1.33T + 59T^{2} \)
61 \( 1 - 12.5iT - 61T^{2} \)
67 \( 1 + (4.71 + 4.71i)T + 67iT^{2} \)
71 \( 1 - 9.22T + 71T^{2} \)
73 \( 1 + (5.82 + 5.82i)T + 73iT^{2} \)
79 \( 1 + 1.91iT - 79T^{2} \)
83 \( 1 + (-8.97 - 8.97i)T + 83iT^{2} \)
89 \( 1 - 4.07T + 89T^{2} \)
97 \( 1 + (-2.69 + 2.69i)T - 97iT^{2} \)
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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

−10.07337612713738497723522910694, −9.085140803331612605856674662435, −8.402835001844483850397436527836, −7.64484258321025066524232529120, −6.65551304358573496035809407186, −5.68169147658784698300195505551, −4.74379585685461947957956247424, −3.60590275343080513751196267881, −2.54447336799137590679711845866, −1.97242494250420357186485827831, 0.20991046025239218846236055541, 2.02453636848500573933264892234, 2.44620021239351776172506185710, 4.36823742442679824883698412859, 5.04814447287998057394200030986, 6.17569762017352061503685932445, 6.74267584339326981923172807453, 7.61595413248912753718320990491, 8.509438504975405135668992458994, 9.029932376550631899206994872130

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