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Properties

Label 2-1470-35.13-c1-0-35
Degree $2$
Conductor $1470$
Sign $-0.966 - 0.256i$
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 + (−1.19 + 1.89i)5-s − 1.00i·6-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (0.493 + 2.18i)10-s − 1.97·11-s + (−0.707 − 0.707i)12-s + (−2.19 + 2.19i)13-s + (0.493 + 2.18i)15-s − 1.00·16-s + (−3.25 − 3.25i)17-s + (−0.707 − 0.707i)18-s − 4.21·19-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.408 − 0.408i)3-s − 0.500i·4-s + (−0.533 + 0.845i)5-s − 0.408i·6-s + (−0.250 − 0.250i)8-s − 0.333i·9-s + (0.156 + 0.689i)10-s − 0.596·11-s + (−0.204 − 0.204i)12-s + (−0.608 + 0.608i)13-s + (0.127 + 0.563i)15-s − 0.250·16-s + (−0.789 − 0.789i)17-s + (−0.166 − 0.166i)18-s − 0.967·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.966 - 0.256i$
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: \((2,\ 1470,\ (\ :1/2),\ -0.966 - 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5726249571\)
\(L(\frac12)\) \(\approx\) \(0.5726249571\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.19 - 1.89i)T \)
7 \( 1 \)
good11 \( 1 + 1.97T + 11T^{2} \)
13 \( 1 + (2.19 - 2.19i)T - 13iT^{2} \)
17 \( 1 + (3.25 + 3.25i)T + 17iT^{2} \)
19 \( 1 + 4.21T + 19T^{2} \)
23 \( 1 + (4.15 + 4.15i)T + 23iT^{2} \)
29 \( 1 + 8.94iT - 29T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + (1.96 - 1.96i)T - 37iT^{2} \)
41 \( 1 + 6.55iT - 41T^{2} \)
43 \( 1 + (6.33 + 6.33i)T + 43iT^{2} \)
47 \( 1 + (-4.29 - 4.29i)T + 47iT^{2} \)
53 \( 1 + (-8.08 - 8.08i)T + 53iT^{2} \)
59 \( 1 - 4.20T + 59T^{2} \)
61 \( 1 - 11.1iT - 61T^{2} \)
67 \( 1 + (-3.89 + 3.89i)T - 67iT^{2} \)
71 \( 1 + 3.86T + 71T^{2} \)
73 \( 1 + (10.7 - 10.7i)T - 73iT^{2} \)
79 \( 1 - 2.55iT - 79T^{2} \)
83 \( 1 + (-9.52 + 9.52i)T - 83iT^{2} \)
89 \( 1 + 6.19T + 89T^{2} \)
97 \( 1 + (1.48 + 1.48i)T + 97iT^{2} \)
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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

−9.054202150436506062872470364920, −8.260700020144303140352589016671, −7.30477793823545914689194375924, −6.73207537652288870012685960094, −5.83356477235334087356049074480, −4.52521819658996519063387489361, −3.93024993443090987342597246561, −2.59734771296872833399298795897, −2.25699491352490323749389898688, −0.16435747898454862091969580807, 1.97503808825808439975101503220, 3.28963088559298613670188566191, 4.12265718178334168312225667111, 4.92341574648798596681671503335, 5.58339973399417929857694686588, 6.72180245951268229481338914913, 7.70581749359504056531567430940, 8.282733647208186762025514956500, 8.867338681196486247025096945187, 9.848957911434826485088817103305

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