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Properties

Label 2-1470-105.104-c1-0-3
Degree $2$
Conductor $1470$
Sign $-0.382 - 0.924i$
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 + (0.321 − 1.70i)3-s + 4-s + (−2.20 + 0.350i)5-s + (0.321 − 1.70i)6-s + 8-s + (−2.79 − 1.09i)9-s + (−2.20 + 0.350i)10-s + 3.78i·11-s + (0.321 − 1.70i)12-s − 6.40·13-s + (−0.113 + 3.87i)15-s + 16-s + 5.57i·17-s + (−2.79 − 1.09i)18-s − 2.10i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.185 − 0.982i)3-s + 0.5·4-s + (−0.987 + 0.156i)5-s + (0.131 − 0.694i)6-s + 0.353·8-s + (−0.930 − 0.365i)9-s + (−0.698 + 0.110i)10-s + 1.14i·11-s + (0.0929 − 0.491i)12-s − 1.77·13-s + (−0.0293 + 0.999i)15-s + 0.250·16-s + 1.35i·17-s + (−0.658 − 0.258i)18-s − 0.482i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.382 - 0.924i$
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: \((2,\ 1470,\ (\ :1/2),\ -0.382 - 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5972924697\)
\(L(\frac12)\) \(\approx\) \(0.5972924697\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.321 + 1.70i)T \)
5 \( 1 + (2.20 - 0.350i)T \)
7 \( 1 \)
good11 \( 1 - 3.78iT - 11T^{2} \)
13 \( 1 + 6.40T + 13T^{2} \)
17 \( 1 - 5.57iT - 17T^{2} \)
19 \( 1 + 2.10iT - 19T^{2} \)
23 \( 1 + 4.70T + 23T^{2} \)
29 \( 1 - 7.55iT - 29T^{2} \)
31 \( 1 + 3.43iT - 31T^{2} \)
37 \( 1 - 2.75iT - 37T^{2} \)
41 \( 1 + 8.77T + 41T^{2} \)
43 \( 1 + 7.04iT - 43T^{2} \)
47 \( 1 + 2.55iT - 47T^{2} \)
53 \( 1 + 3.08T + 53T^{2} \)
59 \( 1 - 12.9T + 59T^{2} \)
61 \( 1 - 10.3iT - 61T^{2} \)
67 \( 1 + 1.48iT - 67T^{2} \)
71 \( 1 - 5.06iT - 71T^{2} \)
73 \( 1 + 3.33T + 73T^{2} \)
79 \( 1 + 7.07T + 79T^{2} \)
83 \( 1 + 9.49iT - 83T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 - 4.90T + 97T^{2} \)
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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

−9.888876119074522106282611785955, −8.647068056188978705086821351004, −7.929933526634007991866248907262, −7.04470987661756033792968899880, −6.92898220682033411525117669770, −5.59398335375417777709531289336, −4.68146448756139492181739619258, −3.78352150661328679538522899699, −2.68121327226735511609904995656, −1.78625187749289016907908837611, 0.16712230474286655572740872056, 2.53510920237876888719454172960, 3.30367891659717422747947154449, 4.19618960420938283621566193168, 4.91676477568885338595349091855, 5.60456414108241564151139448261, 6.78942380411546593021576826771, 7.79584646912537150644507806232, 8.287176107248851550084256843788, 9.419003767462948103031724055949

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