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Properties

Label 2-210-7.5-c2-0-9
Degree $2$
Conductor $210$
Sign $0.103 + 0.994i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 1.22i)2-s + (1.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s + (1.93 + 1.11i)5-s − 2.44i·6-s + (5.10 − 4.79i)7-s − 2.82·8-s + (1.5 − 2.59i)9-s + (2.73 − 1.58i)10-s + (0.919 + 1.59i)11-s + (−2.99 − 1.73i)12-s − 5.40i·13-s + (−2.26 − 9.63i)14-s + 3.87·15-s + (−2.00 + 3.46i)16-s + (8.71 − 5.02i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.5 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s − 0.408i·6-s + (0.728 − 0.684i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (0.273 − 0.158i)10-s + (0.0835 + 0.144i)11-s + (−0.249 − 0.144i)12-s − 0.415i·13-s + (−0.161 − 0.688i)14-s + 0.258·15-s + (−0.125 + 0.216i)16-s + (0.512 − 0.295i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.103 + 0.994i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.103 + 0.994i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.72298 - 1.55294i\)
\(L(\frac12)\) \(\approx\) \(1.72298 - 1.55294i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
3 \( 1 + (-1.5 + 0.866i)T \)
5 \( 1 + (-1.93 - 1.11i)T \)
7 \( 1 + (-5.10 + 4.79i)T \)
good11 \( 1 + (-0.919 - 1.59i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 5.40iT - 169T^{2} \)
17 \( 1 + (-8.71 + 5.02i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (7.96 + 4.59i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-0.460 + 0.797i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 12.5T + 841T^{2} \)
31 \( 1 + (36.1 - 20.8i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (3.64 - 6.30i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 52.3iT - 1.68e3T^{2} \)
43 \( 1 + 8.12T + 1.84e3T^{2} \)
47 \( 1 + (-29.4 - 16.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-52.0 - 90.1i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-12.5 + 7.26i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-20.5 - 11.8i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-7.46 - 12.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 17.9T + 5.04e3T^{2} \)
73 \( 1 + (107. - 62.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-40.4 + 70.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 154. iT - 6.88e3T^{2} \)
89 \( 1 + (-58.9 - 34.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 88.1iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.97763924417380998063981208352, −10.90278689580544019622415408684, −10.16286227359566795383774764080, −9.053651492908161167561698530032, −7.905606467825844605272081804082, −6.84143380173565019352715442682, −5.40679356147192571372529421036, −4.14854185842066092438285550327, −2.78725387599438033204172849071, −1.32873314219706409686147245452, 2.10498006632038874405151502940, 3.79882969927901047404357595070, 5.05206445565510467516714231971, 5.97707465412467643458967092126, 7.37740350400127627603624829971, 8.466613061485183875347145564546, 9.089396780639354044490515236916, 10.29740872466506484862339480236, 11.55847843963443843462378811560, 12.52030718579872585267073101364

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