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Properties

Label 2-1470-35.27-c1-0-9
Degree $2$
Conductor $1470$
Sign $-0.998 + 0.0603i$
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 + (−1.35 + 1.77i)5-s + 1.00i·6-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (−2.21 + 0.300i)10-s + 2.44·11-s + (−0.707 + 0.707i)12-s + (−1.28 − 1.28i)13-s + (−2.21 + 0.300i)15-s − 1.00·16-s + (−5.33 + 5.33i)17-s + (−0.707 + 0.707i)18-s + 0.690·19-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.408 + 0.408i)3-s + 0.500i·4-s + (−0.605 + 0.795i)5-s + 0.408i·6-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (−0.700 + 0.0951i)10-s + 0.738·11-s + (−0.204 + 0.204i)12-s + (−0.356 − 0.356i)13-s + (−0.572 + 0.0776i)15-s − 0.250·16-s + (−1.29 + 1.29i)17-s + (−0.166 + 0.166i)18-s + 0.158·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0603i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0603i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.998 + 0.0603i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ -0.998 + 0.0603i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.582712065\)
\(L(\frac12)\) \(\approx\) \(1.582712065\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.35 - 1.77i)T \)
7 \( 1 \)
good11 \( 1 - 2.44T + 11T^{2} \)
13 \( 1 + (1.28 + 1.28i)T + 13iT^{2} \)
17 \( 1 + (5.33 - 5.33i)T - 17iT^{2} \)
19 \( 1 - 0.690T + 19T^{2} \)
23 \( 1 + (5.65 - 5.65i)T - 23iT^{2} \)
29 \( 1 + 5.08iT - 29T^{2} \)
31 \( 1 - 1.26iT - 31T^{2} \)
37 \( 1 + (-7.16 - 7.16i)T + 37iT^{2} \)
41 \( 1 - 1.15iT - 41T^{2} \)
43 \( 1 + (-0.893 + 0.893i)T - 43iT^{2} \)
47 \( 1 + (4.95 - 4.95i)T - 47iT^{2} \)
53 \( 1 + (-6.74 + 6.74i)T - 53iT^{2} \)
59 \( 1 + 8.06T + 59T^{2} \)
61 \( 1 - 1.09iT - 61T^{2} \)
67 \( 1 + (7.41 + 7.41i)T + 67iT^{2} \)
71 \( 1 - 4.24T + 71T^{2} \)
73 \( 1 + (4.58 + 4.58i)T + 73iT^{2} \)
79 \( 1 + 2.31iT - 79T^{2} \)
83 \( 1 + (7.39 + 7.39i)T + 83iT^{2} \)
89 \( 1 - 16.8T + 89T^{2} \)
97 \( 1 + (4.74 - 4.74i)T - 97iT^{2} \)
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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

−9.896091673520987552511361969127, −9.012827624523748460550225714619, −8.047291549888998601987502086896, −7.64647399749549330549083263862, −6.50293337108311467802804108252, −6.05604952277614070224036837002, −4.66799299042708247503010079628, −3.98651735647156781734873001572, −3.26218580347581250755440996289, −2.09088626014142770332447250235, 0.48313166395244583956521225150, 1.84639241995694315351163587954, 2.86252006076722924345385979129, 4.14611193989415403354547893269, 4.51478698264055296427411559578, 5.68131169542584440055949550876, 6.73780015308313327697835919483, 7.40383446944636209432556231495, 8.484467877181134093921278751022, 9.086716527111734391928314982464

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