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Properties

Label 2-1470-35.27-c1-0-32
Degree $2$
Conductor $1470$
Sign $0.729 - 0.684i$
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 + (1.36 − 1.77i)5-s + 1.00i·6-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (2.21 − 0.286i)10-s + 5.48·11-s + (−0.707 + 0.707i)12-s + (−2.41 − 2.41i)13-s + (2.21 − 0.286i)15-s − 1.00·16-s + (−1.49 + 1.49i)17-s + (−0.707 + 0.707i)18-s + 6.99·19-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.408 + 0.408i)3-s + 0.500i·4-s + (0.610 − 0.791i)5-s + 0.408i·6-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (0.701 − 0.0907i)10-s + 1.65·11-s + (−0.204 + 0.204i)12-s + (−0.670 − 0.670i)13-s + (0.572 − 0.0740i)15-s − 0.250·16-s + (−0.363 + 0.363i)17-s + (−0.166 + 0.166i)18-s + 1.60·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.729 - 0.684i$
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: \((2,\ 1470,\ (\ :1/2),\ 0.729 - 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.069052281\)
\(L(\frac12)\) \(\approx\) \(3.069052281\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.36 + 1.77i)T \)
7 \( 1 \)
good11 \( 1 - 5.48T + 11T^{2} \)
13 \( 1 + (2.41 + 2.41i)T + 13iT^{2} \)
17 \( 1 + (1.49 - 1.49i)T - 17iT^{2} \)
19 \( 1 - 6.99T + 19T^{2} \)
23 \( 1 + (1.24 - 1.24i)T - 23iT^{2} \)
29 \( 1 - 0.684iT - 29T^{2} \)
31 \( 1 + 5.57iT - 31T^{2} \)
37 \( 1 + (-6.92 - 6.92i)T + 37iT^{2} \)
41 \( 1 + 2.50iT - 41T^{2} \)
43 \( 1 + (1.95 - 1.95i)T - 43iT^{2} \)
47 \( 1 + (-2.49 + 2.49i)T - 47iT^{2} \)
53 \( 1 + (6.64 - 6.64i)T - 53iT^{2} \)
59 \( 1 - 10.1T + 59T^{2} \)
61 \( 1 - 1.17iT - 61T^{2} \)
67 \( 1 + (-7.03 - 7.03i)T + 67iT^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + (3.44 + 3.44i)T + 73iT^{2} \)
79 \( 1 - 8.33iT - 79T^{2} \)
83 \( 1 + (4.05 + 4.05i)T + 83iT^{2} \)
89 \( 1 - 7.18T + 89T^{2} \)
97 \( 1 + (13.1 - 13.1i)T - 97iT^{2} \)
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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

−9.580334018449243430886158812888, −8.821996143535585382647108398392, −8.038114243851547920525272629682, −7.16591725002625794116218422520, −6.15531385478969676726828862959, −5.45481194526828655698642468495, −4.56530091299444522298929678536, −3.81954729361858938563240633066, −2.68871631177316362329677258679, −1.28893696787524574377151165677, 1.30580905383877812332609763310, 2.28606054881428928187015830188, 3.22419748046735868633203550199, 4.09959422244087350058925200905, 5.22700090764153503972822851232, 6.28244372941985039611395137685, 6.84087074726459742126838462900, 7.55343412962421809633060617653, 8.954718973566245273954311518198, 9.474788004514914415256791064721

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