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Properties

Label 2-210-35.19-c2-0-8
Degree $2$
Conductor $210$
Sign $0.412 - 0.910i$
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 + (1.34 − 4.81i)5-s + 2.44i·6-s + (−0.180 + 6.99i)7-s + 2.82i·8-s + (−1.5 + 2.59i)9-s + (5.05 − 4.94i)10-s + (7.49 + 12.9i)11-s + (−1.73 + 2.99i)12-s + 12.8·13-s + (−5.16 + 8.44i)14-s + (8.38 − 2.15i)15-s + (−2.00 + 3.46i)16-s + (−10.9 − 18.9i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.288 + 0.5i)3-s + (0.249 + 0.433i)4-s + (0.269 − 0.963i)5-s + 0.408i·6-s + (−0.0258 + 0.999i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.505 − 0.494i)10-s + (0.681 + 1.18i)11-s + (−0.144 + 0.249i)12-s + 0.991·13-s + (−0.369 + 0.603i)14-s + (0.559 − 0.143i)15-s + (−0.125 + 0.216i)16-s + (−0.643 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 - 0.910i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.412 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.412 - 0.910i$
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: \((2,\ 210,\ (\ :1),\ 0.412 - 0.910i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.09397 + 1.35044i\)
\(L(\frac12)\) \(\approx\) \(2.09397 + 1.35044i\)
\(L(2)\) 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 \( 1 + (-1.34 + 4.81i)T \)
7 \( 1 + (0.180 - 6.99i)T \)
good11 \( 1 + (-7.49 - 12.9i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 12.8T + 169T^{2} \)
17 \( 1 + (10.9 + 18.9i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-19.7 - 11.3i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (19.3 + 11.1i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 24.4T + 841T^{2} \)
31 \( 1 + (23.0 - 13.3i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (45.3 + 26.1i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 25.5iT - 1.68e3T^{2} \)
43 \( 1 + 30.7iT - 1.84e3T^{2} \)
47 \( 1 + (-30.3 + 52.6i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-86.4 + 49.9i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (1.24 - 0.716i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-6.52 - 3.76i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (98.9 - 57.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 114.T + 5.04e3T^{2} \)
73 \( 1 + (-11.6 - 20.1i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-31.4 + 54.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 120.T + 6.88e3T^{2} \)
89 \( 1 + (-43.5 - 25.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 46.0T + 9.40e3T^{2} \)
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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

−12.15849141877364268111757449591, −11.88945242618391301602793013431, −10.21453180255992046998229404794, −9.100782878601585414060041695612, −8.605315333327899770592953146946, −7.14418211526693390693639046578, −5.78067089227926411602303886650, −4.93583553585817594990208733890, −3.80673774345269126672909282410, −2.06625476271524877659864831978, 1.36368374252159107646635835087, 3.13421971166577004631935695610, 3.96862330954560880102394663019, 5.95984988494136569523630661107, 6.59980780735227946529773389702, 7.74748487013717027722493768701, 9.044304373777950776880621228241, 10.37910961285884495734699313806, 11.05297651938257806109757214843, 11.85291004413006014093073532304

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