↑

Properties

Label 2-2070-15.2-c1-0-7
Degree $2$
Conductor $2070$
Sign $-0.190 - 0.981i$
Analytic cond. $16.5290$
Root an. cond. $4.06559$
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 − 1.00i·4-s + (1.85 − 1.25i)5-s + (−3.08 − 3.08i)7-s + (0.707 + 0.707i)8-s + (−0.426 + 2.19i)10-s + 1.10i·11-s + (−2.55 + 2.55i)13-s + 4.36·14-s − 1.00·16-s + (−1.10 + 1.10i)17-s + 7.95i·19-s + (−1.25 − 1.85i)20-s + (−0.783 − 0.783i)22-s + (−0.707 − 0.707i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.828 − 0.559i)5-s + (−1.16 − 1.16i)7-s + (0.250 + 0.250i)8-s + (−0.134 + 0.694i)10-s + 0.334i·11-s + (−0.707 + 0.707i)13-s + 1.16·14-s − 0.250·16-s + (−0.268 + 0.268i)17-s + 1.82i·19-s + (−0.279 − 0.414i)20-s + (−0.167 − 0.167i)22-s + (−0.147 − 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.190 - 0.981i$
Analytic conductor: \(16.5290\)
Root analytic conductor: \(4.06559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2070} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :1/2),\ -0.190 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7834654885\)
\(L(\frac12)\) \(\approx\) \(0.7834654885\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-1.85 + 1.25i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (3.08 + 3.08i)T + 7iT^{2} \)
11 \( 1 - 1.10iT - 11T^{2} \)
13 \( 1 + (2.55 - 2.55i)T - 13iT^{2} \)
17 \( 1 + (1.10 - 1.10i)T - 17iT^{2} \)
19 \( 1 - 7.95iT - 19T^{2} \)
29 \( 1 + 6.30T + 29T^{2} \)
31 \( 1 - 3.74T + 31T^{2} \)
37 \( 1 + (-4.56 - 4.56i)T + 37iT^{2} \)
41 \( 1 + 4.15iT - 41T^{2} \)
43 \( 1 + (3.87 - 3.87i)T - 43iT^{2} \)
47 \( 1 + (6.32 - 6.32i)T - 47iT^{2} \)
53 \( 1 + (-3.25 - 3.25i)T + 53iT^{2} \)
59 \( 1 - 12.9T + 59T^{2} \)
61 \( 1 - 13.0T + 61T^{2} \)
67 \( 1 + (-0.0854 - 0.0854i)T + 67iT^{2} \)
71 \( 1 - 2.39iT - 71T^{2} \)
73 \( 1 + (-2.36 + 2.36i)T - 73iT^{2} \)
79 \( 1 - 13.4iT - 79T^{2} \)
83 \( 1 + (4.53 + 4.53i)T + 83iT^{2} \)
89 \( 1 + 1.98T + 89T^{2} \)
97 \( 1 + (-1.85 - 1.85i)T + 97iT^{2} \)
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

−9.622980458101935379026969329895, −8.564836551466262947362962185455, −7.76808484733293590694182047348, −6.90577723096804673943484627933, −6.36288874925814014063300642209, −5.58024930033855532253349918241, −4.50852006258245355901032335976, −3.72039455986492354470404292926, −2.24987098531111090733175963117, −1.14916288762083612869769653397, 0.34673565428515480914458399655, 2.24441100517178709456698451395, 2.69227588336674806858259664058, 3.50947486722955765753604731552, 5.06289639121068821427310756376, 5.74766640755852894118909503893, 6.66130682917359993444807054085, 7.20324364500842178382268389043, 8.443992013108842875314065537764, 9.089981584403686969890619063961

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