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Properties

Label 2-210-105.17-c2-0-25
Degree $2$
Conductor $210$
Sign $-0.377 + 0.925i$
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  + (1.36 + 0.366i)2-s + (−2.51 + 1.63i)3-s + (1.73 + i)4-s + (−2.05 − 4.55i)5-s + (−4.03 + 1.31i)6-s + (−6.80 − 1.63i)7-s + (1.99 + 2i)8-s + (3.65 − 8.22i)9-s + (−1.14 − 6.97i)10-s + (−7.66 − 4.42i)11-s + (−5.99 + 0.315i)12-s + (−5.26 − 5.26i)13-s + (−8.69 − 4.72i)14-s + (12.6 + 8.09i)15-s + (1.99 + 3.46i)16-s + (9.09 − 2.43i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.838 + 0.544i)3-s + (0.433 + 0.250i)4-s + (−0.411 − 0.911i)5-s + (−0.672 + 0.218i)6-s + (−0.972 − 0.233i)7-s + (0.249 + 0.250i)8-s + (0.406 − 0.913i)9-s + (−0.114 − 0.697i)10-s + (−0.696 − 0.402i)11-s + (−0.499 + 0.0262i)12-s + (−0.405 − 0.405i)13-s + (−0.621 − 0.337i)14-s + (0.841 + 0.539i)15-s + (0.124 + 0.216i)16-s + (0.534 − 0.143i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.377 + 0.925i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.377 + 0.925i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.392905 - 0.584763i\)
\(L(\frac12)\) \(\approx\) \(0.392905 - 0.584763i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
3 \( 1 + (2.51 - 1.63i)T \)
5 \( 1 + (2.05 + 4.55i)T \)
7 \( 1 + (6.80 + 1.63i)T \)
good11 \( 1 + (7.66 + 4.42i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (5.26 + 5.26i)T + 169iT^{2} \)
17 \( 1 + (-9.09 + 2.43i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (16.3 + 28.3i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (3.23 - 12.0i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 48.1T + 841T^{2} \)
31 \( 1 + (42.0 + 24.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-4.02 + 15.0i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 45.6T + 1.68e3T^{2} \)
43 \( 1 + (17.0 - 17.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (18.2 - 67.9i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (23.5 - 6.32i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-31.3 - 18.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-41.5 + 23.9i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-20.6 + 5.53i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 42.3iT - 5.04e3T^{2} \)
73 \( 1 + (118. - 31.8i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-53.6 + 31.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-113. + 113. i)T - 6.88e3iT^{2} \)
89 \( 1 + (-104. + 60.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-72.8 + 72.8i)T - 9.40e3iT^{2} \)
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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

−11.98472696578084191904388194297, −11.02711560839129759371364458197, −10.02910130061842921282429528714, −8.965437546021301137476157009036, −7.58093162719410173371426975681, −6.38920472561472765639398841296, −5.33567337897612994384296716073, −4.48006390459887509999349295260, −3.21156463218428150974809940699, −0.33043475516286323521758366945, 2.24726248335512383597289968793, 3.67292854323271627981530385810, 5.17188528906436541381174508023, 6.35780517172689958447464043775, 6.93668128713344512845565149383, 8.123007430557855799543975946253, 10.25069114846226846777930204117, 10.38066818982782758336679715731, 11.84392210745508747073194982606, 12.30550021347848055020429880290

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