↑

Properties

Label 2-210-105.68-c2-0-9
Degree $2$
Conductor $210$
Sign $0.688 + 0.725i$
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 + (−2.49 − 1.66i)3-s + (1.73 − i)4-s + (−4.84 + 1.24i)5-s + (4.01 + 1.35i)6-s + (−0.400 + 6.98i)7-s + (−1.99 + 2i)8-s + (3.46 + 8.30i)9-s + (6.15 − 3.47i)10-s + (4.21 − 2.43i)11-s + (−5.98 − 0.386i)12-s + (13.3 − 13.3i)13-s + (−2.01 − 9.69i)14-s + (14.1 + 4.94i)15-s + (1.99 − 3.46i)16-s + (−25.3 − 6.77i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.832 − 0.554i)3-s + (0.433 − 0.250i)4-s + (−0.968 + 0.249i)5-s + (0.669 + 0.226i)6-s + (−0.0572 + 0.998i)7-s + (−0.249 + 0.250i)8-s + (0.384 + 0.923i)9-s + (0.615 − 0.347i)10-s + (0.382 − 0.221i)11-s + (−0.498 − 0.0321i)12-s + (1.02 − 1.02i)13-s + (−0.143 − 0.692i)14-s + (0.944 + 0.329i)15-s + (0.124 − 0.216i)16-s + (−1.48 − 0.398i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.688 + 0.725i$
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: \((2,\ 210,\ (\ :1),\ 0.688 + 0.725i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.570485 - 0.245054i\)
\(L(\frac12)\) \(\approx\) \(0.570485 - 0.245054i\)
\(L(2)\) 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 \( 1 + (2.49 + 1.66i)T \)
5 \( 1 + (4.84 - 1.24i)T \)
7 \( 1 + (0.400 - 6.98i)T \)
good11 \( 1 + (-4.21 + 2.43i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-13.3 + 13.3i)T - 169iT^{2} \)
17 \( 1 + (25.3 + 6.77i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (-6.44 + 11.1i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (5.04 + 18.8i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 - 41.0T + 841T^{2} \)
31 \( 1 + (-28.4 + 16.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-16.1 - 60.2i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 - 50.2T + 1.68e3T^{2} \)
43 \( 1 + (39.5 + 39.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (0.928 + 3.46i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-55.8 - 14.9i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-41.0 + 23.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (58.7 + 33.9i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-40.3 - 10.7i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 30.7iT - 5.04e3T^{2} \)
73 \( 1 + (-52.4 - 14.0i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (26.6 + 15.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (43.6 + 43.6i)T + 6.88e3iT^{2} \)
89 \( 1 + (54.3 + 31.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-16.7 - 16.7i)T + 9.40e3iT^{2} \)
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

−11.70976181280244025884389632284, −11.25006422261559614490545332667, −10.25947370813690732704471709064, −8.692186787168617583126973483754, −8.136842106819828843755199191332, −6.80256354149670698842213541651, −6.13729621409091076939463829647, −4.71296706190501626304595169580, −2.70677020035373568716684992201, −0.63016130238795948751941342151, 1.06903779430786466854238270030, 3.80299050558035365148845100949, 4.43586132731628184373518175723, 6.32825843671588100665886562550, 7.14130846358268773091623524319, 8.433834350938909611214113307398, 9.391476686522111349850750176612, 10.49836723494842363909554090079, 11.24236069228148592498581768928, 11.81733013666863184676936141868

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