Properties

Label 2-1470-35.13-c1-0-22
Degree $2$
Conductor $1470$
Sign $0.930 + 0.365i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + (1.84 − 1.26i)5-s + 1.00i·6-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (−0.410 + 2.19i)10-s + 5.16·11-s + (−0.707 − 0.707i)12-s + (−0.184 + 0.184i)13-s + (0.410 − 2.19i)15-s − 1.00·16-s + (0.750 + 0.750i)17-s + (0.707 + 0.707i)18-s + 4.08·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.408 − 0.408i)3-s − 0.500i·4-s + (0.824 − 0.565i)5-s + 0.408i·6-s + (0.250 + 0.250i)8-s − 0.333i·9-s + (−0.129 + 0.695i)10-s + 1.55·11-s + (−0.204 − 0.204i)12-s + (−0.0512 + 0.0512i)13-s + (0.106 − 0.567i)15-s − 0.250·16-s + (0.182 + 0.182i)17-s + (0.166 + 0.166i)18-s + 0.937·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.930 + 0.365i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (1273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.930 + 0.365i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.960907219\)
\(L(\frac12)\) \(\approx\) \(1.960907219\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.84 + 1.26i)T \)
7 \( 1 \)
good11 \( 1 - 5.16T + 11T^{2} \)
13 \( 1 + (0.184 - 0.184i)T - 13iT^{2} \)
17 \( 1 + (-0.750 - 0.750i)T + 17iT^{2} \)
19 \( 1 - 4.08T + 19T^{2} \)
23 \( 1 + (2.36 + 2.36i)T + 23iT^{2} \)
29 \( 1 + 0.387iT - 29T^{2} \)
31 \( 1 - 10.6iT - 31T^{2} \)
37 \( 1 + (-2.51 + 2.51i)T - 37iT^{2} \)
41 \( 1 - 7.05iT - 41T^{2} \)
43 \( 1 + (2.13 + 2.13i)T + 43iT^{2} \)
47 \( 1 + (7.37 + 7.37i)T + 47iT^{2} \)
53 \( 1 + (-7.15 - 7.15i)T + 53iT^{2} \)
59 \( 1 + 9.70T + 59T^{2} \)
61 \( 1 + 11.0iT - 61T^{2} \)
67 \( 1 + (0.0355 - 0.0355i)T - 67iT^{2} \)
71 \( 1 - 15.7T + 71T^{2} \)
73 \( 1 + (-3.22 + 3.22i)T - 73iT^{2} \)
79 \( 1 + 7.46iT - 79T^{2} \)
83 \( 1 + (-0.409 + 0.409i)T - 83iT^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 + (1.32 + 1.32i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.377288774477209778661461839491, −8.674698992637338541335599552676, −8.025502222903773197314942774909, −6.89315532534619891011504002028, −6.43729645235570031662164534307, −5.51375787648337863069534242414, −4.54486459067485056532241918655, −3.31257999946253393164826675208, −1.87277796171372457621078023635, −1.07379590188401452319889477509, 1.31709004389374278911818663096, 2.38020569454196036848873740058, 3.38936716403777943822727951241, 4.16619967559218030471683576775, 5.46130300092162467879099405659, 6.37285231659484692678966164415, 7.23129457897842087174500593898, 8.092230508794671629645193152253, 9.113076581806004904874620167394, 9.645120128960034884206416794294

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