Properties

Label 2-210-35.19-c2-0-13
Degree $2$
Conductor $210$
Sign $-0.883 + 0.468i$
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 + (−0.214 − 4.99i)5-s + 2.44i·6-s + (5.88 − 3.78i)7-s − 2.82i·8-s + (−1.5 + 2.59i)9-s + (−3.27 + 6.26i)10-s + (−0.746 − 1.29i)11-s + (1.73 − 2.99i)12-s + 4.37·13-s + (−9.88 + 0.479i)14-s + (−7.30 + 4.64i)15-s + (−2.00 + 3.46i)16-s + (−7.58 − 13.1i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.288 − 0.5i)3-s + (0.249 + 0.433i)4-s + (−0.0428 − 0.999i)5-s + 0.408i·6-s + (0.840 − 0.541i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.327 + 0.626i)10-s + (−0.0678 − 0.117i)11-s + (0.144 − 0.249i)12-s + 0.336·13-s + (−0.706 + 0.0342i)14-s + (−0.487 + 0.309i)15-s + (−0.125 + 0.216i)16-s + (−0.446 − 0.773i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.883 + 0.468i$
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: \((2,\ 210,\ (\ :1),\ -0.883 + 0.468i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.212186 - 0.852043i\)
\(L(\frac12)\) \(\approx\) \(0.212186 - 0.852043i\)
\(L(2)\) 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 \( 1 + (0.214 + 4.99i)T \)
7 \( 1 + (-5.88 + 3.78i)T \)
good11 \( 1 + (0.746 + 1.29i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 4.37T + 169T^{2} \)
17 \( 1 + (7.58 + 13.1i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (0.814 + 0.469i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (33.3 + 19.2i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 4.01T + 841T^{2} \)
31 \( 1 + (12.0 - 6.94i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (22.7 + 13.1i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 29.2iT - 1.68e3T^{2} \)
43 \( 1 - 30.8iT - 1.84e3T^{2} \)
47 \( 1 + (-41.6 + 72.1i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (32.8 - 18.9i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-72.0 + 41.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-24.5 - 14.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-63.4 + 36.6i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 83.1T + 5.04e3T^{2} \)
73 \( 1 + (-66.4 - 115. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-26.8 + 46.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 77.1T + 6.88e3T^{2} \)
89 \( 1 + (-41.9 - 24.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 37.5T + 9.40e3T^{2} \)
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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

−11.69192982192149203191544390374, −10.89088414982761693749482081609, −9.774215317407105227357439564120, −8.560191031609795003478360049884, −7.980012281791378757374179370560, −6.79530361832045755430092407921, −5.32087037470132051246596539609, −4.11650009375924980854729278338, −1.99076566069825766686672307751, −0.62183014235507690222865199229, 2.08515288168656816758755777164, 3.89573750844089292083505988165, 5.46641800302445249470146060966, 6.37219695380011511526129647561, 7.62077776656179771992860557361, 8.540898300065313081334522823054, 9.687630793804171395733077319013, 10.63119132763480653000332534353, 11.26578049102658590456418684065, 12.19636414592959197743966730407

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