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Properties

Label 2-1470-21.20-c1-0-44
Degree $2$
Conductor $1470$
Sign $0.953 + 0.300i$
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  + i·2-s + (1.47 − 0.907i)3-s − 4-s + 5-s + (0.907 + 1.47i)6-s − i·8-s + (1.35 − 2.67i)9-s + i·10-s + 3.28i·11-s + (−1.47 + 0.907i)12-s − 5.91i·13-s + (1.47 − 0.907i)15-s + 16-s + 2.40·17-s + (2.67 + 1.35i)18-s − 5.51i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.851 − 0.523i)3-s − 0.5·4-s + 0.447·5-s + (0.370 + 0.602i)6-s − 0.353i·8-s + (0.451 − 0.892i)9-s + 0.316i·10-s + 0.990i·11-s + (−0.425 + 0.261i)12-s − 1.64i·13-s + (0.380 − 0.234i)15-s + 0.250·16-s + 0.583·17-s + (0.631 + 0.319i)18-s − 1.26i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.953 + 0.300i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.953 + 0.300i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.337869775\)
\(L(\frac12)\) \(\approx\) \(2.337869775\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-1.47 + 0.907i)T \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 - 3.28iT - 11T^{2} \)
13 \( 1 + 5.91iT - 13T^{2} \)
17 \( 1 - 2.40T + 17T^{2} \)
19 \( 1 + 5.51iT - 19T^{2} \)
23 \( 1 + 6.49iT - 23T^{2} \)
29 \( 1 - 3.80iT - 29T^{2} \)
31 \( 1 - 5.20iT - 31T^{2} \)
37 \( 1 + 1.88T + 37T^{2} \)
41 \( 1 - 0.103T + 41T^{2} \)
43 \( 1 + 1.48T + 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 - 2.43iT - 53T^{2} \)
59 \( 1 - 3.65T + 59T^{2} \)
61 \( 1 - 14.2iT - 61T^{2} \)
67 \( 1 + 3.35T + 67T^{2} \)
71 \( 1 + 13.9iT - 71T^{2} \)
73 \( 1 + 8.11iT - 73T^{2} \)
79 \( 1 - 9.15T + 79T^{2} \)
83 \( 1 - 6.54T + 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 - 7.90iT - 97T^{2} \)
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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

−9.143682702250456845987089431494, −8.666593053395217027519360002675, −7.72845391268848349991904976818, −7.17491369258151525800478997421, −6.40760061348931447876935253461, −5.39232299900925038859340270985, −4.54681808399393462039199458656, −3.26345164800160429812787449808, −2.40417992199608154893987806286, −0.915409968293358413307248886132, 1.50205421128954073922254511978, 2.40020331873221140082459814399, 3.58574893250887079530713298626, 4.05768089876888469825904419479, 5.27077445223523839087120313950, 6.08617478908117804731797158279, 7.36666619141205287482019504489, 8.214615313126091056311788889736, 8.942309694953460479480571855764, 9.647109723711054229643544376875

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