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Properties

Label 2-1470-35.9-c1-0-35
Degree $2$
Conductor $1470$
Sign $-0.720 + 0.693i$
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  + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (1.40 − 1.74i)5-s − 0.999·6-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−2.08 + 0.806i)10-s + (−3.05 − 5.28i)11-s + (0.866 + 0.499i)12-s + 1.68i·13-s + (0.344 − 2.20i)15-s + (−0.5 + 0.866i)16-s + (5.92 − 3.41i)17-s + (−0.866 + 0.499i)18-s + (−0.844 + 1.46i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (0.627 − 0.778i)5-s − 0.408·6-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.659 + 0.255i)10-s + (−0.920 − 1.59i)11-s + (0.249 + 0.144i)12-s + 0.468i·13-s + (0.0888 − 0.570i)15-s + (−0.125 + 0.216i)16-s + (1.43 − 0.829i)17-s + (−0.204 + 0.117i)18-s + (−0.193 + 0.335i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.720 + 0.693i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ -0.720 + 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.397804856\)
\(L(\frac12)\) \(\approx\) \(1.397804856\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-1.40 + 1.74i)T \)
7 \( 1 \)
good11 \( 1 + (3.05 + 5.28i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.68iT - 13T^{2} \)
17 \( 1 + (-5.92 + 3.41i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.844 - 1.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.00 - 1.15i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.41T + 29T^{2} \)
31 \( 1 + (0.344 + 0.596i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.00 + 1.15i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.14T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + (2.65 + 1.53i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.05 + 5.23i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.52 - 6.09i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.73 + 8.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.5 - 6.10i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-3.46 + 2i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.65 + 6.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.5iT - 83T^{2} \)
89 \( 1 + (-6.10 + 10.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.95iT - 97T^{2} \)
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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

−9.201988456373317197794377195879, −8.437760165879998754489091018986, −7.924549251983415375496713321432, −6.99374307188712801776414556350, −5.78492115503279560514532923415, −5.24782692511969884233955268864, −3.73592081807713678209434476694, −2.89040383699476133190527646754, −1.76766799789996355645738969578, −0.62636661183455996920048601623, 1.71964066673957649844321829414, 2.59271051550768479648945727519, 3.66995599126216221776534408483, 5.04950925394087301910890522581, 5.69123586083441579462922726794, 6.83522622089615697249503669751, 7.47922993207963938798800359894, 8.078309224627240163512592795212, 9.153946881056251976506063552464, 9.850392908998420940575651715562

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