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Properties

Label 2-1470-35.27-c1-0-14
Degree $2$
Conductor $1470$
Sign $0.892 + 0.451i$
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 + (−1.84 − 1.26i)5-s + 1.00i·6-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (0.410 + 2.19i)10-s + 5.16·11-s + (0.707 − 0.707i)12-s + (0.184 + 0.184i)13-s + (0.410 + 2.19i)15-s − 1.00·16-s + (−0.750 + 0.750i)17-s + (0.707 − 0.707i)18-s − 4.08·19-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (−0.824 − 0.565i)5-s + 0.408i·6-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (0.129 + 0.695i)10-s + 1.55·11-s + (0.204 − 0.204i)12-s + (0.0512 + 0.0512i)13-s + (0.106 + 0.567i)15-s − 0.250·16-s + (−0.182 + 0.182i)17-s + (0.166 − 0.166i)18-s − 0.937·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.892 + 0.451i$
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: \((2,\ 1470,\ (\ :1/2),\ 0.892 + 0.451i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8843416355\)
\(L(\frac12)\) \(\approx\) \(0.8843416355\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.84 + 1.26i)T \)
7 \( 1 \)
good11 \( 1 - 5.16T + 11T^{2} \)
13 \( 1 + (-0.184 - 0.184i)T + 13iT^{2} \)
17 \( 1 + (0.750 - 0.750i)T - 17iT^{2} \)
19 \( 1 + 4.08T + 19T^{2} \)
23 \( 1 + (2.36 - 2.36i)T - 23iT^{2} \)
29 \( 1 - 0.387iT - 29T^{2} \)
31 \( 1 - 10.6iT - 31T^{2} \)
37 \( 1 + (-2.51 - 2.51i)T + 37iT^{2} \)
41 \( 1 - 7.05iT - 41T^{2} \)
43 \( 1 + (2.13 - 2.13i)T - 43iT^{2} \)
47 \( 1 + (-7.37 + 7.37i)T - 47iT^{2} \)
53 \( 1 + (-7.15 + 7.15i)T - 53iT^{2} \)
59 \( 1 - 9.70T + 59T^{2} \)
61 \( 1 + 11.0iT - 61T^{2} \)
67 \( 1 + (0.0355 + 0.0355i)T + 67iT^{2} \)
71 \( 1 - 15.7T + 71T^{2} \)
73 \( 1 + (3.22 + 3.22i)T + 73iT^{2} \)
79 \( 1 - 7.46iT - 79T^{2} \)
83 \( 1 + (0.409 + 0.409i)T + 83iT^{2} \)
89 \( 1 - 14.1T + 89T^{2} \)
97 \( 1 + (-1.32 + 1.32i)T - 97iT^{2} \)
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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

−9.347511518414048057827860664716, −8.603815632352681587454988184791, −8.095242264618718521820843613958, −6.97292418112321095318184373455, −6.49018940997294901544618067221, −5.17110579215407567907108312031, −4.18622165466673857610539942582, −3.47900227680175994371075815816, −1.88859689746743394462673614177, −0.899265460258512422010505912375, 0.62793602423771340582673137973, 2.36507230750464786925223988298, 4.00466718910151633575823508376, 4.20836723089094945584677761259, 5.70622080842397765676351530426, 6.42312547731797826240446998314, 7.07270820015044396325750952612, 7.942033392827959747072192531560, 8.825219888415518219489412974740, 9.428537370659390419033818841494

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