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Properties

Label 2-1470-21.20-c1-0-4
Degree $2$
Conductor $1470$
Sign $-0.474 - 0.880i$
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 + (−1.05 + 1.37i)3-s − 4-s + 5-s + (1.37 + 1.05i)6-s + i·8-s + (−0.772 − 2.89i)9-s − i·10-s − 1.09i·11-s + (1.05 − 1.37i)12-s + 3.66i·13-s + (−1.05 + 1.37i)15-s + 16-s − 4.39·17-s + (−2.89 + 0.772i)18-s + 4.59i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.609 + 0.792i)3-s − 0.5·4-s + 0.447·5-s + (0.560 + 0.430i)6-s + 0.353i·8-s + (−0.257 − 0.966i)9-s − 0.316i·10-s − 0.331i·11-s + (0.304 − 0.396i)12-s + 1.01i·13-s + (−0.272 + 0.354i)15-s + 0.250·16-s − 1.06·17-s + (−0.683 + 0.182i)18-s + 1.05i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 - 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.474 - 0.880i$
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: \((2,\ 1470,\ (\ :1/2),\ -0.474 - 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5701376295\)
\(L(\frac12)\) \(\approx\) \(0.5701376295\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.05 - 1.37i)T \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 + 1.09iT - 11T^{2} \)
13 \( 1 - 3.66iT - 13T^{2} \)
17 \( 1 + 4.39T + 17T^{2} \)
19 \( 1 - 4.59iT - 19T^{2} \)
23 \( 1 + 1.51iT - 23T^{2} \)
29 \( 1 - 1.85iT - 29T^{2} \)
31 \( 1 + 4.34iT - 31T^{2} \)
37 \( 1 + 11.7T + 37T^{2} \)
41 \( 1 - 6.08T + 41T^{2} \)
43 \( 1 + 1.89T + 43T^{2} \)
47 \( 1 - 5.13T + 47T^{2} \)
53 \( 1 - 11.5iT - 53T^{2} \)
59 \( 1 + 8.82T + 59T^{2} \)
61 \( 1 - 11.4iT - 61T^{2} \)
67 \( 1 + 12.2T + 67T^{2} \)
71 \( 1 + 2.24iT - 71T^{2} \)
73 \( 1 - 6.69iT - 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 - 1.22T + 89T^{2} \)
97 \( 1 - 8.60iT - 97T^{2} \)
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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

−9.927411853215734175351730016466, −9.041228735145176112293642074079, −8.709021864801390638491840113150, −7.25615584184812456177888815921, −6.23597146988528589270131106427, −5.59729134209793105776742849986, −4.50981916495886675856440384562, −3.96282569549535491368725434107, −2.76617584451379793407557818562, −1.50434866652249950616568581282, 0.24527398768188051633735302853, 1.75811237987401814591786857119, 2.99294510315668763933245448588, 4.56554901273007505417220284095, 5.27599500866086789939851350279, 6.03292201440504322554687614998, 6.87018926574215141325409458161, 7.35360323643582707605013005467, 8.358667608127677927626107924624, 9.000123825705135447296810077081

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