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Properties

Label 2-210-35.19-c2-0-14
Degree $2$
Conductor $210$
Sign $-0.979 - 0.202i$
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  + (−1.22 − 0.707i)2-s + (0.866 + 1.5i)3-s + (0.999 + 1.73i)4-s + (−2.13 − 4.52i)5-s − 2.44i·6-s + (−6.07 − 3.47i)7-s − 2.82i·8-s + (−1.5 + 2.59i)9-s + (−0.578 + 7.04i)10-s + (10.3 + 17.9i)11-s + (−1.73 + 2.99i)12-s − 23.2·13-s + (4.99 + 8.54i)14-s + (4.92 − 7.12i)15-s + (−2.00 + 3.46i)16-s + (−3.75 − 6.49i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.288 + 0.5i)3-s + (0.249 + 0.433i)4-s + (−0.427 − 0.904i)5-s − 0.408i·6-s + (−0.868 − 0.495i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.0578 + 0.704i)10-s + (0.940 + 1.62i)11-s + (−0.144 + 0.249i)12-s − 1.78·13-s + (0.356 + 0.610i)14-s + (0.328 − 0.474i)15-s + (−0.125 + 0.216i)16-s + (−0.220 − 0.382i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.979 - 0.202i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.979 - 0.202i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00338897 + 0.0331272i\)
\(L(\frac12)\) \(\approx\) \(0.00338897 + 0.0331272i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 + (2.13 + 4.52i)T \)
7 \( 1 + (6.07 + 3.47i)T \)
good11 \( 1 + (-10.3 - 17.9i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 23.2T + 169T^{2} \)
17 \( 1 + (3.75 + 6.49i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (24.6 + 14.2i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (4.24 + 2.45i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 36.3T + 841T^{2} \)
31 \( 1 + (-1.34 + 0.777i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-8.75 - 5.05i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 3.17iT - 1.68e3T^{2} \)
43 \( 1 + 17.2iT - 1.84e3T^{2} \)
47 \( 1 + (5.77 - 9.99i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-33.1 + 19.1i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-51.4 + 29.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (26.3 + 15.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (103. - 59.4i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 32.6T + 5.04e3T^{2} \)
73 \( 1 + (-25.4 - 44.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-68.3 + 118. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 86.0T + 6.88e3T^{2} \)
89 \( 1 + (17.8 + 10.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 17.1T + 9.40e3T^{2} \)
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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

−11.71290749523448257749437291484, −10.33113389786290023906288509781, −9.517480283258131011909098117295, −9.077744474927649510585089677353, −7.61629165467828350507423282745, −6.85519153408767972740837895411, −4.78619754919205697271789137382, −3.99199520754879766921501153970, −2.22054410343300945746370299557, −0.02023836238385141352972918504, 2.39232331122140382486726667473, 3.67696166799244780283378629845, 5.92918795826071819113362583688, 6.59947674962691186970761916575, 7.62294314623350662398251054336, 8.634375083057505415318682172572, 9.565686883090216517197393298026, 10.63440764411814388851416131223, 11.66712796692238824273435381627, 12.50664393526718456006264489551

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