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Properties

Label 2-1470-105.104-c1-0-79
Degree $2$
Conductor $1470$
Sign $-0.0961 + 0.995i$
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 + (1.17 − 1.27i)3-s + 4-s + (1.07 − 1.95i)5-s + (1.17 − 1.27i)6-s + 8-s + (−0.259 − 2.98i)9-s + (1.07 − 1.95i)10-s − 6.01i·11-s + (1.17 − 1.27i)12-s + 0.864·13-s + (−1.23 − 3.67i)15-s + 16-s + 2.84i·17-s + (−0.259 − 2.98i)18-s + 7.12i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.675 − 0.737i)3-s + 0.5·4-s + (0.482 − 0.875i)5-s + (0.477 − 0.521i)6-s + 0.353·8-s + (−0.0865 − 0.996i)9-s + (0.341 − 0.619i)10-s − 1.81i·11-s + (0.337 − 0.368i)12-s + 0.239·13-s + (−0.319 − 0.947i)15-s + 0.250·16-s + 0.690i·17-s + (−0.0612 − 0.704i)18-s + 1.63i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0961 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0961 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.0961 + 0.995i$
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: \((2,\ 1470,\ (\ :1/2),\ -0.0961 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.540811125\)
\(L(\frac12)\) \(\approx\) \(3.540811125\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.17 + 1.27i)T \)
5 \( 1 + (-1.07 + 1.95i)T \)
7 \( 1 \)
good11 \( 1 + 6.01iT - 11T^{2} \)
13 \( 1 - 0.864T + 13T^{2} \)
17 \( 1 - 2.84iT - 17T^{2} \)
19 \( 1 - 7.12iT - 19T^{2} \)
23 \( 1 + 2.10T + 23T^{2} \)
29 \( 1 - 9.40iT - 29T^{2} \)
31 \( 1 - 2.68iT - 31T^{2} \)
37 \( 1 + 6.81iT - 37T^{2} \)
41 \( 1 + 5.32T + 41T^{2} \)
43 \( 1 - 2.68iT - 43T^{2} \)
47 \( 1 + 3.12iT - 47T^{2} \)
53 \( 1 - 11.9T + 53T^{2} \)
59 \( 1 - 7.09T + 59T^{2} \)
61 \( 1 - 9.33iT - 61T^{2} \)
67 \( 1 + 6.56iT - 67T^{2} \)
71 \( 1 + 4.46iT - 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 - 3.35T + 79T^{2} \)
83 \( 1 + 1.28iT - 83T^{2} \)
89 \( 1 - 3.08T + 89T^{2} \)
97 \( 1 - 1.90T + 97T^{2} \)
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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

−8.927398129072962451108035196535, −8.504420272338920255062668281950, −7.83657735133102093317644679848, −6.66034742761996281308367088604, −5.88937756440235308939165936594, −5.40450726563355042605165190656, −3.88516650924362863874502877558, −3.35428162924359105191886044003, −2.01219130338311626296183060273, −1.06971434291776467317747543347, 2.16892989516656224000830206204, 2.60623383886220529701700339623, 3.79554848707113500466404341279, 4.61006190492546255778679408381, 5.32484653239048950759882802811, 6.55703977738523344330912391171, 7.17147709181745155962481108908, 7.964471332847529706736914678938, 9.181396943160811944584900332667, 9.871081608564539930886883515113

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