Properties

Label 2-1470-35.13-c1-0-5
Degree $2$
Conductor $1470$
Sign $-0.386 - 0.922i$
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 − 2i)5-s − 1.00i·6-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (0.707 + 2.12i)10-s − 1.96·11-s + (0.707 + 0.707i)12-s + (−2.37 + 2.37i)13-s + (0.707 + 2.12i)15-s − 1.00·16-s + (3.38 + 3.38i)17-s + (0.707 + 0.707i)18-s − 3.41·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.408 + 0.408i)3-s − 0.500i·4-s + (0.447 − 0.894i)5-s − 0.408i·6-s + (0.250 + 0.250i)8-s − 0.333i·9-s + (0.223 + 0.670i)10-s − 0.592·11-s + (0.204 + 0.204i)12-s + (−0.659 + 0.659i)13-s + (0.182 + 0.547i)15-s − 0.250·16-s + (0.821 + 0.821i)17-s + (0.166 + 0.166i)18-s − 0.783·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.386 - 0.922i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (1273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7829143313\)
\(L(\frac12)\) \(\approx\) \(0.7829143313\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2i)T \)
7 \( 1 \)
good11 \( 1 + 1.96T + 11T^{2} \)
13 \( 1 + (2.37 - 2.37i)T - 13iT^{2} \)
17 \( 1 + (-3.38 - 3.38i)T + 17iT^{2} \)
19 \( 1 + 3.41T + 19T^{2} \)
23 \( 1 + (3 + 3i)T + 23iT^{2} \)
29 \( 1 - 4.22iT - 29T^{2} \)
31 \( 1 - 1.98iT - 31T^{2} \)
37 \( 1 + (-5.35 + 5.35i)T - 37iT^{2} \)
41 \( 1 - 11.5iT - 41T^{2} \)
43 \( 1 + (-7.81 - 7.81i)T + 43iT^{2} \)
47 \( 1 + (-4.93 - 4.93i)T + 47iT^{2} \)
53 \( 1 + (-8.39 - 8.39i)T + 53iT^{2} \)
59 \( 1 + 4.19T + 59T^{2} \)
61 \( 1 - 3.92iT - 61T^{2} \)
67 \( 1 + (-2.36 + 2.36i)T - 67iT^{2} \)
71 \( 1 + 4.14T + 71T^{2} \)
73 \( 1 + (-2.01 + 2.01i)T - 73iT^{2} \)
79 \( 1 + 11.5iT - 79T^{2} \)
83 \( 1 + (10.9 - 10.9i)T - 83iT^{2} \)
89 \( 1 - 6.02T + 89T^{2} \)
97 \( 1 + (-5.08 - 5.08i)T + 97iT^{2} \)
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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

−9.639727594672841156915862267972, −9.036757876612941604264420818099, −8.205494403151110263160542026571, −7.49206023241535713159686679611, −6.26963658740201419522906232269, −5.80612722809718605037710976265, −4.80409881380762323069046942937, −4.20135381231065571145168943142, −2.48185236196017427737921236479, −1.17667895589225222122478501363, 0.42344643220803674872216500629, 2.09427694380215730711449944056, 2.74862303573892604528683585210, 3.91704895813570101330885335653, 5.31541873509206917962655732532, 5.93290841794273400539724063996, 7.10863674835007771619040429757, 7.52297165488946719099865839435, 8.389577092380421429978189185423, 9.552586529634863323177085231259

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