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Properties

Label 2-1470-35.13-c1-0-29
Degree $2$
Conductor $1470$
Sign $-0.991 + 0.131i$
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 + 1.51·11-s + (0.707 + 0.707i)12-s + (−3.93 + 3.93i)13-s + (−0.707 − 2.12i)15-s − 1.00·16-s + (−3.07 − 3.07i)17-s + (−0.707 − 0.707i)18-s + 0.585·19-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 + 0.457·11-s + (0.204 + 0.204i)12-s + (−1.09 + 1.09i)13-s + (−0.182 − 0.547i)15-s − 0.250·16-s + (−0.745 − 0.745i)17-s + (−0.166 − 0.166i)18-s + 0.134·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.991 + 0.131i$
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: \((2,\ 1470,\ (\ :1/2),\ -0.991 + 0.131i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.09809430261\)
\(L(\frac12)\) \(\approx\) \(0.09809430261\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.51T + 11T^{2} \)
13 \( 1 + (3.93 - 3.93i)T - 13iT^{2} \)
17 \( 1 + (3.07 + 3.07i)T + 17iT^{2} \)
19 \( 1 - 0.585T + 19T^{2} \)
23 \( 1 + (3 + 3i)T + 23iT^{2} \)
29 \( 1 - 4.24iT - 29T^{2} \)
31 \( 1 + 10.4iT - 31T^{2} \)
37 \( 1 + (-1.55 + 1.55i)T - 37iT^{2} \)
41 \( 1 - 6.68iT - 41T^{2} \)
43 \( 1 + (3.83 + 3.83i)T + 43iT^{2} \)
47 \( 1 + (3.97 + 3.97i)T + 47iT^{2} \)
53 \( 1 + (7.02 + 7.02i)T + 53iT^{2} \)
59 \( 1 - 0.729T + 59T^{2} \)
61 \( 1 - 3.03iT - 61T^{2} \)
67 \( 1 + (9.93 - 9.93i)T - 67iT^{2} \)
71 \( 1 + 5.70T + 71T^{2} \)
73 \( 1 + (-6.48 + 6.48i)T - 73iT^{2} \)
79 \( 1 - 6.68iT - 79T^{2} \)
83 \( 1 + (10.1 - 10.1i)T - 83iT^{2} \)
89 \( 1 + 18.0T + 89T^{2} \)
97 \( 1 + (12.2 + 12.2i)T + 97iT^{2} \)
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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

−9.581898272976506307110792931347, −8.398374658265788976807202533384, −7.15841619321995694988909368347, −6.71176649188343064995576944960, −5.76403508664844628700243753762, −4.57956831032329880496631294993, −4.15451129132226810728996530491, −2.99786166935462887640167568606, −2.05447587307490452273699255040, −0.03323709178928975713406723550, 1.58411726175159512112311555231, 3.07976111185575996624659788712, 4.23022983542264964278993787030, 4.96589625845074001871055896138, 5.71528573220100040037468643931, 6.57808650138917220111191378006, 7.52476535769578556479775061291, 8.059684857436676466764668310513, 8.870669697820393298001470552473, 9.808084188170362666487076898092

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