↑

Properties

Label 2-210-7.2-c3-0-1
Degree $2$
Conductor $210$
Sign $-0.997 - 0.0767i$
Analytic cond. $12.3904$
Root an. cond. $3.52000$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (1.5 + 2.59i)3-s + (−1.99 − 3.46i)4-s + (2.5 − 4.33i)5-s − 6·6-s + (−2.58 + 18.3i)7-s + 7.99·8-s + (−4.5 + 7.79i)9-s + (5 + 8.66i)10-s + (−3.91 − 6.77i)11-s + (6.00 − 10.3i)12-s − 32.8·13-s + (−29.1 − 22.8i)14-s + 15.0·15-s + (−8 + 13.8i)16-s + (34.4 + 59.6i)17-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 + (−0.139 + 0.990i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 + 0.273i)10-s + (−0.107 − 0.185i)11-s + (0.144 − 0.249i)12-s − 0.700·13-s + (−0.556 − 0.435i)14-s + 0.258·15-s + (−0.125 + 0.216i)16-s + (0.491 + 0.851i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0767i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.997 - 0.0767i$
Analytic conductor: \(12.3904\)
Root analytic conductor: \(3.52000\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :3/2),\ -0.997 - 0.0767i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0357321 + 0.929938i\)
\(L(\frac12)\) \(\approx\) \(0.0357321 + 0.929938i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
3 \( 1 + (-1.5 - 2.59i)T \)
5 \( 1 + (-2.5 + 4.33i)T \)
7 \( 1 + (2.58 - 18.3i)T \)
good11 \( 1 + (3.91 + 6.77i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 32.8T + 2.19e3T^{2} \)
17 \( 1 + (-34.4 - 59.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (22.8 - 39.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (108. - 188. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 126.T + 2.43e4T^{2} \)
31 \( 1 + (84.5 + 146. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-121. + 210. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 286.T + 6.89e4T^{2} \)
43 \( 1 + 54.2T + 7.95e4T^{2} \)
47 \( 1 + (113. - 196. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-26.8 - 46.5i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-435. - 753. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-232. + 402. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-105. - 182. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 236.T + 3.57e5T^{2} \)
73 \( 1 + (-176. - 305. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (391. - 677. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 956.T + 5.71e5T^{2} \)
89 \( 1 + (93.0 - 161. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 10.1T + 9.12e5T^{2} \)
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

−12.44926197338647358025334461894, −11.36591621180435090092839046234, −9.998192771849248882535002298754, −9.443802148732140131768285673507, −8.447297496274576150933458403723, −7.60156812747363580014502578183, −5.95771697167499847451237349937, −5.33966141220376589050668163248, −3.79477233171524835756394076682, −1.99826675736112997550102117907, 0.40822855465790036132351985741, 2.10339035321407649469355062346, 3.36025165098572157238674815081, 4.79634137783997384358518690071, 6.61916243599681186708255808427, 7.40550340846519955359493379448, 8.444757614506127608378020084009, 9.733965007949919295577958511199, 10.33656575364361046338514559006, 11.41998221344457437223856641601

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