Properties

Label 2-1470-105.104-c1-0-28
Degree $2$
Conductor $1470$
Sign $0.639 - 0.769i$
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  + 2-s + (−1.67 + 0.425i)3-s + 4-s + (2.18 − 0.461i)5-s + (−1.67 + 0.425i)6-s + 8-s + (2.63 − 1.42i)9-s + (2.18 − 0.461i)10-s + 4.08i·11-s + (−1.67 + 0.425i)12-s − 3.50·13-s + (−3.47 + 1.70i)15-s + 16-s − 0.437i·17-s + (2.63 − 1.42i)18-s + 7.69i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.969 + 0.245i)3-s + 0.5·4-s + (0.978 − 0.206i)5-s + (−0.685 + 0.173i)6-s + 0.353·8-s + (0.879 − 0.475i)9-s + (0.691 − 0.145i)10-s + 1.23i·11-s + (−0.484 + 0.122i)12-s − 0.970·13-s + (−0.897 + 0.440i)15-s + 0.250·16-s − 0.106i·17-s + (0.621 − 0.336i)18-s + 1.76i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.639 - 0.769i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (1469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.639 - 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.235100119\)
\(L(\frac12)\) \(\approx\) \(2.235100119\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.67 - 0.425i)T \)
5 \( 1 + (-2.18 + 0.461i)T \)
7 \( 1 \)
good11 \( 1 - 4.08iT - 11T^{2} \)
13 \( 1 + 3.50T + 13T^{2} \)
17 \( 1 + 0.437iT - 17T^{2} \)
19 \( 1 - 7.69iT - 19T^{2} \)
23 \( 1 - 7.39T + 23T^{2} \)
29 \( 1 - 4.95iT - 29T^{2} \)
31 \( 1 + 6.81iT - 31T^{2} \)
37 \( 1 + 4.51iT - 37T^{2} \)
41 \( 1 - 1.34T + 41T^{2} \)
43 \( 1 - 6.67iT - 43T^{2} \)
47 \( 1 + 2.41iT - 47T^{2} \)
53 \( 1 - 0.424T + 53T^{2} \)
59 \( 1 - 2.75T + 59T^{2} \)
61 \( 1 - 5.75iT - 61T^{2} \)
67 \( 1 - 11.3iT - 67T^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 - 9.06T + 73T^{2} \)
79 \( 1 - 3.37T + 79T^{2} \)
83 \( 1 - 17.1iT - 83T^{2} \)
89 \( 1 + 4.52T + 89T^{2} \)
97 \( 1 - 7.93T + 97T^{2} \)
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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

−9.854286626323409133799957305755, −9.155059644300777055376672115249, −7.65497852810201549485589753214, −6.96422928502194265683906995583, −6.16345364234276967803762999367, −5.32762575693909694177679432911, −4.85424557284577886755679659070, −3.89933649657287004673433859713, −2.45124840416675328765969181116, −1.39127585039797886952106887815, 0.859921581255463906971822246500, 2.29887743096985291050328980231, 3.21496611237070067181117626435, 4.80089291603869089386978164817, 5.14717802566426540827866838031, 6.07627767784185094397780960684, 6.73766233926046764227576338461, 7.33648722034441218146211186725, 8.665809798833932559144843593412, 9.536373662298719501548372031975

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