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Properties

Label 2-1470-21.20-c1-0-51
Degree $2$
Conductor $1470$
Sign $-0.895 - 0.445i$
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  − i·2-s + (−0.431 − 1.67i)3-s − 4-s + 5-s + (−1.67 + 0.431i)6-s + i·8-s + (−2.62 + 1.44i)9-s − i·10-s − 5.13i·11-s + (0.431 + 1.67i)12-s − 5.00i·13-s + (−0.431 − 1.67i)15-s + 16-s + 3.75·17-s + (1.44 + 2.62i)18-s + 2.69i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.249 − 0.968i)3-s − 0.5·4-s + 0.447·5-s + (−0.684 + 0.176i)6-s + 0.353i·8-s + (−0.875 + 0.482i)9-s − 0.316i·10-s − 1.54i·11-s + (0.124 + 0.484i)12-s − 1.38i·13-s + (−0.111 − 0.433i)15-s + 0.250·16-s + 0.911·17-s + (0.341 + 0.619i)18-s + 0.617i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.895 - 0.445i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ -0.895 - 0.445i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.153908475\)
\(L(\frac12)\) \(\approx\) \(1.153908475\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.431 + 1.67i)T \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 + 5.13iT - 11T^{2} \)
13 \( 1 + 5.00iT - 13T^{2} \)
17 \( 1 - 3.75T + 17T^{2} \)
19 \( 1 - 2.69iT - 19T^{2} \)
23 \( 1 + 2.48iT - 23T^{2} \)
29 \( 1 + 6.18iT - 29T^{2} \)
31 \( 1 - 4.77iT - 31T^{2} \)
37 \( 1 - 0.0524T + 37T^{2} \)
41 \( 1 + 6.55T + 41T^{2} \)
43 \( 1 + 9.45T + 43T^{2} \)
47 \( 1 - 3.06T + 47T^{2} \)
53 \( 1 + 1.11iT - 53T^{2} \)
59 \( 1 + 13.2T + 59T^{2} \)
61 \( 1 - 4.14iT - 61T^{2} \)
67 \( 1 + 0.896T + 67T^{2} \)
71 \( 1 + 13.4iT - 71T^{2} \)
73 \( 1 - 6.98iT - 73T^{2} \)
79 \( 1 + 16.7T + 79T^{2} \)
83 \( 1 - 1.37T + 83T^{2} \)
89 \( 1 - 5.34T + 89T^{2} \)
97 \( 1 - 0.633iT - 97T^{2} \)
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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

−8.913523151828306656139983818756, −8.238290277098952244088013673384, −7.68774954985627977701055737330, −6.36094158183110077897557988385, −5.77515778923124702497578554704, −5.07892991165054514673636071628, −3.42325813317504811116948502851, −2.84157028858722391421340602431, −1.51514938147101399274001064306, −0.48720810523080815623943865486, 1.77699444150545387269981763677, 3.28035606449325370808713509738, 4.40743013544648445413526437926, 4.90251984279524696319345970963, 5.79368153403451782924157342296, 6.74064012491342243300947764362, 7.34652031342167261630492152528, 8.535927061202141052816228965358, 9.329341169558657546724110451277, 9.743899580027140370703734918530

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