Properties

Label 2-1470-35.27-c1-0-3
Degree $2$
Conductor $1470$
Sign $0.254 - 0.966i$
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 + (−2.10 + 0.741i)5-s + 1.00i·6-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (2.01 + 0.967i)10-s + 1.16·11-s + (0.707 − 0.707i)12-s + (−3.61 − 3.61i)13-s + (2.01 + 0.967i)15-s − 1.00·16-s + (4.48 − 4.48i)17-s + (0.707 − 0.707i)18-s − 2.15·19-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (−0.943 + 0.331i)5-s + 0.408i·6-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (0.637 + 0.305i)10-s + 0.352·11-s + (0.204 − 0.204i)12-s + (−1.00 − 1.00i)13-s + (0.520 + 0.249i)15-s − 0.250·16-s + (1.08 − 1.08i)17-s + (0.166 − 0.166i)18-s − 0.494·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.254 - 0.966i$
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: \((2,\ 1470,\ (\ :1/2),\ 0.254 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3178308874\)
\(L(\frac12)\) \(\approx\) \(0.3178308874\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.10 - 0.741i)T \)
7 \( 1 \)
good11 \( 1 - 1.16T + 11T^{2} \)
13 \( 1 + (3.61 + 3.61i)T + 13iT^{2} \)
17 \( 1 + (-4.48 + 4.48i)T - 17iT^{2} \)
19 \( 1 + 2.15T + 19T^{2} \)
23 \( 1 + (1.80 - 1.80i)T - 23iT^{2} \)
29 \( 1 - 7.29iT - 29T^{2} \)
31 \( 1 + 3.14iT - 31T^{2} \)
37 \( 1 + (2.88 + 2.88i)T + 37iT^{2} \)
41 \( 1 - 4.21iT - 41T^{2} \)
43 \( 1 + (-2.87 + 2.87i)T - 43iT^{2} \)
47 \( 1 + (7.84 - 7.84i)T - 47iT^{2} \)
53 \( 1 + (7.17 - 7.17i)T - 53iT^{2} \)
59 \( 1 + 2.85T + 59T^{2} \)
61 \( 1 - 10.0iT - 61T^{2} \)
67 \( 1 + (-10.3 - 10.3i)T + 67iT^{2} \)
71 \( 1 + 6.87T + 71T^{2} \)
73 \( 1 + (3.69 + 3.69i)T + 73iT^{2} \)
79 \( 1 + 0.468iT - 79T^{2} \)
83 \( 1 + (-9.87 - 9.87i)T + 83iT^{2} \)
89 \( 1 + 2.12T + 89T^{2} \)
97 \( 1 + (-6.30 + 6.30i)T - 97iT^{2} \)
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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

−9.801945473956604295596138020801, −8.894183128306567267161294305095, −7.78853692202852054316702702744, −7.57317854348727049943188907032, −6.68705008104554789997870479171, −5.50068408896293028233249939955, −4.58264946192388947319171804261, −3.40763314740498856191357695511, −2.63064254746715797768476109125, −1.06826749352163522023034670040, 0.18979948993923302925246470038, 1.79002430893774364614199623360, 3.52504968752672673796057337827, 4.39319872769365371435008885236, 5.11364356309204236578355737869, 6.22711455999357205927978503267, 6.89760538303167903878872607370, 7.88685490266691678303605926673, 8.403846221701883971475002594101, 9.362362707201684508479716543353

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