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Properties

Label 2-210-35.23-c2-0-7
Degree $2$
Conductor $210$
Sign $-0.186 - 0.982i$
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 + (−0.448 + 1.67i)3-s + (−1.73 + i)4-s + (4.99 + 0.232i)5-s − 2.44·6-s + (6.85 − 1.39i)7-s + (−2 − 1.99i)8-s + (−2.59 − 1.50i)9-s + (1.51 + 6.90i)10-s + (6.05 + 10.4i)11-s + (−0.896 − 3.34i)12-s + (12.6 + 12.6i)13-s + (4.42 + 8.85i)14-s + (−2.62 + 8.25i)15-s + (1.99 − 3.46i)16-s + (−16.4 − 4.41i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.149 + 0.557i)3-s + (−0.433 + 0.250i)4-s + (0.998 + 0.0464i)5-s − 0.408·6-s + (0.979 − 0.199i)7-s + (−0.250 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.151 + 0.690i)10-s + (0.550 + 0.953i)11-s + (−0.0747 − 0.278i)12-s + (0.971 + 0.971i)13-s + (0.315 + 0.632i)14-s + (−0.175 + 0.550i)15-s + (0.124 − 0.216i)16-s + (−0.968 − 0.259i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.186 - 0.982i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.186 - 0.982i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.23244 + 1.48903i\)
\(L(\frac12)\) \(\approx\) \(1.23244 + 1.48903i\)
\(L(2)\) 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 \( 1 + (0.448 - 1.67i)T \)
5 \( 1 + (-4.99 - 0.232i)T \)
7 \( 1 + (-6.85 + 1.39i)T \)
good11 \( 1 + (-6.05 - 10.4i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-12.6 - 12.6i)T + 169iT^{2} \)
17 \( 1 + (16.4 + 4.41i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (30.0 + 17.3i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (4.04 - 1.08i)T + (458. - 264.5i)T^{2} \)
29 \( 1 - 30.5iT - 841T^{2} \)
31 \( 1 + (-11.3 - 19.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (10.9 + 40.8i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 - 68.1T + 1.68e3T^{2} \)
43 \( 1 + (48.6 + 48.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (-15.6 - 58.4i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-10.1 + 38.0i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-50.9 + 29.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-18.7 + 32.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (109. + 29.4i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 27.4T + 5.04e3T^{2} \)
73 \( 1 + (-33.4 + 125. i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (9.99 + 5.77i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (11.4 + 11.4i)T + 6.88e3iT^{2} \)
89 \( 1 + (29.3 + 16.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (26.7 - 26.7i)T - 9.40e3iT^{2} \)
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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

−12.58004564418040004594079327868, −11.25075052678910403269988513269, −10.52215081119164784868252720351, −9.131738330142944252469464033759, −8.754407489190539454677867290135, −7.03847942829335423119481641706, −6.28019605441105314511493430484, −4.91893310073717505947241958081, −4.19976215645840416452117878895, −1.95205427920113062310385271721, 1.22713533517475452257480073993, 2.43852719957260066159114338932, 4.20102089121430995635392475333, 5.72981704085834466772305735560, 6.28322718049192892265479228535, 8.245727798581849835870872307096, 8.725052119769109763029957133774, 10.22469283192087365250645852487, 10.98906077428606525362079657999, 11.77788159941348651315127349196

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