↑

Properties

Label 2-210-105.68-c2-0-1
Degree $2$
Conductor $210$
Sign $-0.850 - 0.526i$
Analytic cond. $5.72208$
Root an. cond. $2.39208$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 + 0.366i)2-s + (−2.99 + 0.157i)3-s + (1.73 − i)4-s + (2.05 − 4.55i)5-s + (4.03 − 1.31i)6-s + (−6.80 + 1.63i)7-s + (−1.99 + 2i)8-s + (8.95 − 0.945i)9-s + (−1.14 + 6.97i)10-s + (7.66 − 4.42i)11-s + (−5.03 + 3.26i)12-s + (−5.26 + 5.26i)13-s + (8.69 − 4.72i)14-s + (−5.44 + 13.9i)15-s + (1.99 − 3.46i)16-s + (−9.09 − 2.43i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.998 + 0.0525i)3-s + (0.433 − 0.250i)4-s + (0.411 − 0.911i)5-s + (0.672 − 0.218i)6-s + (−0.972 + 0.233i)7-s + (−0.249 + 0.250i)8-s + (0.994 − 0.105i)9-s + (−0.114 + 0.697i)10-s + (0.696 − 0.402i)11-s + (−0.419 + 0.272i)12-s + (−0.405 + 0.405i)13-s + (0.621 − 0.337i)14-s + (−0.363 + 0.931i)15-s + (0.124 − 0.216i)16-s + (−0.534 − 0.143i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.850 - 0.526i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.850 - 0.526i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0429669 + 0.151003i\)
\(L(\frac12)\) \(\approx\) \(0.0429669 + 0.151003i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
3 \( 1 + (2.99 - 0.157i)T \)
5 \( 1 + (-2.05 + 4.55i)T \)
7 \( 1 + (6.80 - 1.63i)T \)
good11 \( 1 + (-7.66 + 4.42i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (5.26 - 5.26i)T - 169iT^{2} \)
17 \( 1 + (9.09 + 2.43i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (16.3 - 28.3i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-3.23 - 12.0i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + 48.1T + 841T^{2} \)
31 \( 1 + (42.0 - 24.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-4.02 - 15.0i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 - 45.6T + 1.68e3T^{2} \)
43 \( 1 + (17.0 + 17.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (-18.2 - 67.9i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-23.5 - 6.32i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (31.3 - 18.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-41.5 - 23.9i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-20.6 - 5.53i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 42.3iT - 5.04e3T^{2} \)
73 \( 1 + (118. + 31.8i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-53.6 - 31.0i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (113. + 113. i)T + 6.88e3iT^{2} \)
89 \( 1 + (104. + 60.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-72.8 - 72.8i)T + 9.40e3iT^{2} \)
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

−12.45871948184560084419391573059, −11.55792219475300213610947074755, −10.46455695846715025831550817764, −9.490878807904127896409220964051, −8.928899368695613044945961770819, −7.38054487373436891677815477312, −6.23327669659264766238652882641, −5.59288179753727820633327210147, −4.03581415688882346017761851071, −1.62816109637419878205495961299, 0.12038259933104606863489759727, 2.27541392127793534040597389036, 3.97580870467100602537368117719, 5.76558670964054706795916669794, 6.80099158452437556915664027616, 7.23676567200838977724701317706, 9.169023858934232292107647857670, 9.870056065498766409527723361237, 10.83520429069014610121064286938, 11.34835216039724589281916205100

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