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Properties

Label 2-210-105.38-c2-0-2
Degree $2$
Conductor $210$
Sign $-0.996 - 0.0836i$
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 + (2.84 + 0.941i)3-s + (−1.73 − i)4-s + (−4.99 + 0.262i)5-s + (−2.32 + 3.54i)6-s + (−4.50 + 5.35i)7-s + (2 − 1.99i)8-s + (7.22 + 5.36i)9-s + (1.46 − 6.91i)10-s + (−8.15 − 4.70i)11-s + (−3.99 − 4.47i)12-s + (−16.1 + 16.1i)13-s + (−5.66 − 8.11i)14-s + (−14.4 − 3.95i)15-s + (1.99 + 3.46i)16-s + (−2.42 − 9.03i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.949 + 0.313i)3-s + (−0.433 − 0.250i)4-s + (−0.998 + 0.0525i)5-s + (−0.388 + 0.591i)6-s + (−0.643 + 0.765i)7-s + (0.250 − 0.249i)8-s + (0.802 + 0.596i)9-s + (0.146 − 0.691i)10-s + (−0.741 − 0.428i)11-s + (−0.332 − 0.373i)12-s + (−1.24 + 1.24i)13-s + (−0.404 − 0.579i)14-s + (−0.964 − 0.263i)15-s + (0.124 + 0.216i)16-s + (−0.142 − 0.531i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0836i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.996 - 0.0836i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.996 - 0.0836i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0374470 + 0.894016i\)
\(L(\frac12)\) \(\approx\) \(0.0374470 + 0.894016i\)
\(L(2)\) 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 \( 1 + (-2.84 - 0.941i)T \)
5 \( 1 + (4.99 - 0.262i)T \)
7 \( 1 + (4.50 - 5.35i)T \)
good11 \( 1 + (8.15 + 4.70i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (16.1 - 16.1i)T - 169iT^{2} \)
17 \( 1 + (2.42 + 9.03i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (-13.1 - 22.8i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (23.0 + 6.16i)T + (458. + 264.5i)T^{2} \)
29 \( 1 - 34.0T + 841T^{2} \)
31 \( 1 + (-2.88 - 1.66i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (46.2 + 12.3i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 3.31T + 1.68e3T^{2} \)
43 \( 1 + (-28.9 - 28.9i)T + 1.84e3iT^{2} \)
47 \( 1 + (-3.09 - 0.830i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-15.2 - 56.7i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-33.5 - 19.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-80.5 + 46.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (0.981 + 3.66i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 26.4iT - 5.04e3T^{2} \)
73 \( 1 + (-7.31 - 27.3i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (38.5 - 22.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-70.6 - 70.6i)T + 6.88e3iT^{2} \)
89 \( 1 + (118. - 68.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-118. - 118. i)T + 9.40e3iT^{2} \)
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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

−12.58036822716190954921873016760, −11.85862932427247337042022951216, −10.29083573730251163926063935631, −9.469674780266014158329258654396, −8.525718793155182747037422084317, −7.73138766263745127831940595343, −6.77610935493900710566898288733, −5.17935276531775674837500048724, −3.97375525384775947203961732876, −2.61996953240788910507213191574, 0.46210030053690816511972865881, 2.63379506862620845203304542774, 3.59612561071094885216609113417, 4.82672028858125484471876050529, 7.07364079385500871291029901479, 7.69523254689838827325985518485, 8.613838006942810749459941682528, 9.936812671820294689731549110516, 10.40371798919515308910222816423, 11.88133089484024301013750285426

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