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Properties

Label 2-1470-21.20-c1-0-41
Degree $2$
Conductor $1470$
Sign $0.999 - 0.0339i$
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 + (1.17 + 1.27i)3-s − 4-s − 5-s + (−1.27 + 1.17i)6-s − i·8-s + (−0.226 + 2.99i)9-s − i·10-s − 4.95i·11-s + (−1.17 − 1.27i)12-s − 6.37i·13-s + (−1.17 − 1.27i)15-s + 16-s + 3.62·17-s + (−2.99 − 0.226i)18-s − 4.20i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.679 + 0.733i)3-s − 0.5·4-s − 0.447·5-s + (−0.518 + 0.480i)6-s − 0.353i·8-s + (−0.0753 + 0.997i)9-s − 0.316i·10-s − 1.49i·11-s + (−0.339 − 0.366i)12-s − 1.76i·13-s + (−0.304 − 0.327i)15-s + 0.250·16-s + 0.879·17-s + (−0.705 − 0.0532i)18-s − 0.965i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.999 - 0.0339i$
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: \((2,\ 1470,\ (\ :1/2),\ 0.999 - 0.0339i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.594072295\)
\(L(\frac12)\) \(\approx\) \(1.594072295\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.17 - 1.27i)T \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + 4.95iT - 11T^{2} \)
13 \( 1 + 6.37iT - 13T^{2} \)
17 \( 1 - 3.62T + 17T^{2} \)
19 \( 1 + 4.20iT - 19T^{2} \)
23 \( 1 + 1.91iT - 23T^{2} \)
29 \( 1 + 0.800iT - 29T^{2} \)
31 \( 1 + 5.03iT - 31T^{2} \)
37 \( 1 - 5.82T + 37T^{2} \)
41 \( 1 + 7.45T + 41T^{2} \)
43 \( 1 + 4.64T + 43T^{2} \)
47 \( 1 - 0.607T + 47T^{2} \)
53 \( 1 - 14.3iT - 53T^{2} \)
59 \( 1 + 1.75T + 59T^{2} \)
61 \( 1 - 3.42iT - 61T^{2} \)
67 \( 1 - 7.98T + 67T^{2} \)
71 \( 1 - 2.48iT - 71T^{2} \)
73 \( 1 + 15.0iT - 73T^{2} \)
79 \( 1 - 2.66T + 79T^{2} \)
83 \( 1 + 7.27T + 83T^{2} \)
89 \( 1 - 1.88T + 89T^{2} \)
97 \( 1 + 17.6iT - 97T^{2} \)
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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

−9.330701057669312213825564935818, −8.486783523865132668778705281139, −8.062480153945201231363465343670, −7.38951726606563800228882410970, −6.04750256341900829062979852975, −5.41877854184919021431301920288, −4.49577883151147421153865885316, −3.40007487886948598318973998802, −2.89153189468290162025094063883, −0.61597473886796335931874729934, 1.45974156017425311635189245567, 2.11693476085179191267262842339, 3.41252908809086011837528744717, 4.12198214836046686294914464636, 5.12182921945043882218443310300, 6.54669730967386297839757631862, 7.14438301486061943065559947823, 7.985282674844474610361789444833, 8.705061263539228350402951821839, 9.685651065708251420474030475901

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