↑

Properties

Label 2-1470-7.2-c1-0-17
Degree $2$
Conductor $1470$
Sign $0.827 + 0.561i$
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  + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − 0.999·6-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 + 0.866i)10-s + (−1.70 − 2.95i)11-s + (0.499 − 0.866i)12-s + 0.999·15-s + (−0.5 + 0.866i)16-s + (−0.707 − 1.22i)17-s + (−0.499 − 0.866i)18-s + (1.41 − 2.44i)19-s − 0.999·20-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s − 0.408·6-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 + 0.273i)10-s + (−0.514 − 0.891i)11-s + (0.144 − 0.249i)12-s + 0.258·15-s + (−0.125 + 0.216i)16-s + (−0.171 − 0.297i)17-s + (−0.117 − 0.204i)18-s + (0.324 − 0.561i)19-s − 0.223·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.183668133\)
\(L(\frac12)\) \(\approx\) \(1.183668133\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good11 \( 1 + (1.70 + 2.95i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (0.707 + 1.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.41 + 2.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.414 + 0.717i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.242T + 29T^{2} \)
31 \( 1 + (4.53 + 7.85i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.707 - 1.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.17T + 41T^{2} \)
43 \( 1 - 7.41T + 43T^{2} \)
47 \( 1 + (-2.53 + 4.39i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.65 + 11.5i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.24 + 12.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.171 - 0.297i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.94 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.17T + 71T^{2} \)
73 \( 1 + (1.82 + 3.16i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.65 - 9.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 + (-5.24 + 9.08i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.17T + 97T^{2} \)
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

−9.377741144118413939796833008656, −8.621086309442217681683081684942, −7.980680836433718063210783897643, −7.12810469369161009240860317043, −6.04838804013094700368736240085, −5.36919440661996387875593296889, −4.54036799671397139120768088094, −3.43270072598801718893289850816, −2.22719185993947789928075511618, −0.52611030048546418460169902068, 1.39027138690386347039920585880, 2.35356893214594785948174154581, 3.25413934576986458118547600675, 4.33942250385754095274347934415, 5.45568883557095004276687322825, 6.47493504717400522196668687869, 7.45342189021668347562444862163, 7.83636492918261762718205637999, 9.014196874026487006450353350191, 9.463545801076700489384994801030

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