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Properties

Label 2-1470-21.20-c1-0-25
Degree $2$
Conductor $1470$
Sign $0.818 - 0.574i$
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 + (−1.68 − 0.421i)3-s − 4-s + 5-s + (0.421 − 1.68i)6-s − i·8-s + (2.64 + 1.41i)9-s + i·10-s − 0.193i·11-s + (1.68 + 0.421i)12-s + 1.54i·13-s + (−1.68 − 0.421i)15-s + 16-s + 0.529·17-s + (−1.41 + 2.64i)18-s − 6.38i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.969 − 0.243i)3-s − 0.5·4-s + 0.447·5-s + (0.171 − 0.685i)6-s − 0.353i·8-s + (0.881 + 0.471i)9-s + 0.316i·10-s − 0.0584i·11-s + (0.484 + 0.121i)12-s + 0.429i·13-s + (−0.433 − 0.108i)15-s + 0.250·16-s + 0.128·17-s + (−0.333 + 0.623i)18-s − 1.46i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.818 - 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.818 - 0.574i$
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: \((2,\ 1470,\ (\ :1/2),\ 0.818 - 0.574i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.202593939\)
\(L(\frac12)\) \(\approx\) \(1.202593939\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 + 0.421i)T \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 + 0.193iT - 11T^{2} \)
13 \( 1 - 1.54iT - 13T^{2} \)
17 \( 1 - 0.529T + 17T^{2} \)
19 \( 1 + 6.38iT - 19T^{2} \)
23 \( 1 - 4.24iT - 23T^{2} \)
29 \( 1 - 4.87iT - 29T^{2} \)
31 \( 1 + 9.26iT - 31T^{2} \)
37 \( 1 - 1.76T + 37T^{2} \)
41 \( 1 + 9.91T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 - 9.80T + 47T^{2} \)
53 \( 1 - 0.0649iT - 53T^{2} \)
59 \( 1 - 12.0T + 59T^{2} \)
61 \( 1 - 4.28iT - 61T^{2} \)
67 \( 1 - 4.83T + 67T^{2} \)
71 \( 1 + 6.29iT - 71T^{2} \)
73 \( 1 - 8.09iT - 73T^{2} \)
79 \( 1 - 6.77T + 79T^{2} \)
83 \( 1 - 2.11T + 83T^{2} \)
89 \( 1 - 16.3T + 89T^{2} \)
97 \( 1 + 8.24iT - 97T^{2} \)
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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

−9.485860781627895607616020293115, −8.858928704248877420135369365756, −7.65422238296901011203858954998, −7.07220574829641769957073329210, −6.31167645566591191619927433060, −5.54947172162955139592639797180, −4.87645734006657877522086358612, −3.89234149967634239506286980416, −2.27786477421979932925818383642, −0.819106759248205317239594598234, 0.859963272605215467133856013695, 2.08698804546234449090758530506, 3.43377656945471239384205302491, 4.34327951057960132312286113910, 5.28481059825245925011277855828, 5.93454000279986519247406570540, 6.82044639004862066536918313717, 7.910158453957506871750171183818, 8.836766445874152484171728323639, 9.765659705759964019676876941836

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