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Properties

Label 2-210-105.47-c2-0-16
Degree $2$
Conductor $210$
Sign $0.541 + 0.840i$
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  + (−0.366 − 1.36i)2-s + (−2.80 − 1.05i)3-s + (−1.73 + i)4-s + (4.87 − 1.11i)5-s + (−0.414 + 4.22i)6-s + (0.132 + 6.99i)7-s + (2 + 1.99i)8-s + (6.77 + 5.92i)9-s + (−3.30 − 6.25i)10-s + (9.30 − 5.36i)11-s + (5.91 − 0.979i)12-s + (3.28 + 3.28i)13-s + (9.51 − 2.74i)14-s + (−14.8 − 2.02i)15-s + (1.99 − 3.46i)16-s + (4.29 − 16.0i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.936 − 0.351i)3-s + (−0.433 + 0.250i)4-s + (0.974 − 0.222i)5-s + (−0.0690 + 0.703i)6-s + (0.0189 + 0.999i)7-s + (0.250 + 0.249i)8-s + (0.752 + 0.658i)9-s + (−0.330 − 0.625i)10-s + (0.845 − 0.488i)11-s + (0.493 − 0.0816i)12-s + (0.252 + 0.252i)13-s + (0.679 − 0.195i)14-s + (−0.990 − 0.135i)15-s + (0.124 − 0.216i)16-s + (0.252 − 0.942i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.541 + 0.840i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.541 + 0.840i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.09568 - 0.597661i\)
\(L(\frac12)\) \(\approx\) \(1.09568 - 0.597661i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
3 \( 1 + (2.80 + 1.05i)T \)
5 \( 1 + (-4.87 + 1.11i)T \)
7 \( 1 + (-0.132 - 6.99i)T \)
good11 \( 1 + (-9.30 + 5.36i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-3.28 - 3.28i)T + 169iT^{2} \)
17 \( 1 + (-4.29 + 16.0i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (-1.04 + 1.81i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (12.7 - 3.40i)T + (458. - 264.5i)T^{2} \)
29 \( 1 - 21.8T + 841T^{2} \)
31 \( 1 + (-40.3 + 23.3i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-40.7 + 10.9i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 41.7T + 1.68e3T^{2} \)
43 \( 1 + (32.4 - 32.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (67.3 - 18.0i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (8.36 - 31.2i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-8.21 + 4.74i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-100. - 58.3i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-0.604 + 2.25i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 82.6iT - 5.04e3T^{2} \)
73 \( 1 + (-23.8 + 89.0i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-27.5 - 15.9i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-37.0 + 37.0i)T - 6.88e3iT^{2} \)
89 \( 1 + (136. + 78.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-41.8 + 41.8i)T - 9.40e3iT^{2} \)
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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

−11.78564200340296888731898266565, −11.34277051372994059755947064838, −9.983618198234596655117344528069, −9.336922324031452575264689252711, −8.181053609362260494154675107375, −6.51911448470442748871169344402, −5.75099676646852520949250901372, −4.59922370458303845941547250232, −2.56125618605938326603777441014, −1.14349601886575375701511314298, 1.22054014876384500671020700502, 3.92291617680405871430481491682, 5.06576446338068562430775061654, 6.33088734878803561750588737689, 6.75036853277173532634826213007, 8.222530002545189451071651849731, 9.715007709332480034810415256681, 10.12153933303956692970156647895, 11.07138516180478697963089623974, 12.35889417361568169070785152966

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