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Properties

Label 2-1470-35.4-c1-0-9
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: $\chi_{1470} (949, \cdot )$
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

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

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