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Properties

Label 2-210-105.47-c2-0-17
Degree $2$
Conductor $210$
Sign $0.175 + 0.984i$
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 + (−2.99 + 0.197i)3-s + (−1.73 + i)4-s + (−3.67 + 3.38i)5-s + (1.36 + 4.01i)6-s + (6.78 − 1.72i)7-s + (2 + 1.99i)8-s + (8.92 − 1.18i)9-s + (5.97 + 3.78i)10-s + (−5.25 + 3.03i)11-s + (4.98 − 3.33i)12-s + (−4.95 − 4.95i)13-s + (−4.83 − 8.63i)14-s + (10.3 − 10.8i)15-s + (1.99 − 3.46i)16-s + (6.62 − 24.7i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.997 + 0.0657i)3-s + (−0.433 + 0.250i)4-s + (−0.735 + 0.677i)5-s + (0.227 + 0.669i)6-s + (0.969 − 0.246i)7-s + (0.250 + 0.249i)8-s + (0.991 − 0.131i)9-s + (0.597 + 0.378i)10-s + (−0.477 + 0.275i)11-s + (0.415 − 0.277i)12-s + (−0.381 − 0.381i)13-s + (−0.345 − 0.616i)14-s + (0.689 − 0.724i)15-s + (0.124 − 0.216i)16-s + (0.389 − 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.175 + 0.984i$
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: \((2,\ 210,\ (\ :1),\ 0.175 + 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.636891 - 0.533495i\)
\(L(\frac12)\) \(\approx\) \(0.636891 - 0.533495i\)
\(L(2)\) 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 \( 1 + (2.99 - 0.197i)T \)
5 \( 1 + (3.67 - 3.38i)T \)
7 \( 1 + (-6.78 + 1.72i)T \)
good11 \( 1 + (5.25 - 3.03i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (4.95 + 4.95i)T + 169iT^{2} \)
17 \( 1 + (-6.62 + 24.7i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (-8.47 + 14.6i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-27.7 + 7.44i)T + (458. - 264.5i)T^{2} \)
29 \( 1 - 29.7T + 841T^{2} \)
31 \( 1 + (36.5 - 21.1i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-46.2 + 12.4i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 2.99T + 1.68e3T^{2} \)
43 \( 1 + (-22.2 + 22.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-61.1 + 16.3i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (0.456 - 1.70i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (72.6 - 41.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (18.0 + 10.4i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (8.61 - 32.1i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 10.6iT - 5.04e3T^{2} \)
73 \( 1 + (-25.3 + 94.5i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (74.6 + 43.1i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (47.0 - 47.0i)T - 6.88e3iT^{2} \)
89 \( 1 + (-138. - 80.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-81.7 + 81.7i)T - 9.40e3iT^{2} \)
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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

−11.75078507529265090733535571795, −10.95440662773279381899226496266, −10.46750459058020338769341678804, −9.190242266383216488509537385599, −7.63572905806017798588142196460, −7.11195675390976014709076047492, −5.25268485769967077981899096232, −4.46225522584565155560177188172, −2.81237827187010003083085379195, −0.69608142972849980648578409674, 1.22195537578852900703241507545, 4.16490606458373437138759789784, 5.12352688942628561721559538611, 5.98520436090343810373254962730, 7.47688907459918845234048261842, 8.072539966670642594908855217338, 9.246758503054329766054044843907, 10.57805452290671458038563565870, 11.39682806897146360191101146339, 12.34037854962313892211884763834

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