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Properties

Label 2-210-105.17-c2-0-0
Degree $2$
Conductor $210$
Sign $-0.947 + 0.318i$
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 + (−0.197 + 2.99i)3-s + (1.73 + i)4-s + (−1.09 − 4.87i)5-s + (1.36 − 4.01i)6-s + (−1.72 + 6.78i)7-s + (−1.99 − 2i)8-s + (−8.92 − 1.18i)9-s + (−0.293 + 7.06i)10-s + (−5.25 − 3.03i)11-s + (−3.33 + 4.98i)12-s + (4.95 + 4.95i)13-s + (4.83 − 8.63i)14-s + (14.8 − 2.30i)15-s + (1.99 + 3.46i)16-s + (−24.7 + 6.62i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.0657 + 0.997i)3-s + (0.433 + 0.250i)4-s + (−0.218 − 0.975i)5-s + (0.227 − 0.669i)6-s + (−0.246 + 0.969i)7-s + (−0.249 − 0.250i)8-s + (−0.991 − 0.131i)9-s + (−0.0293 + 0.706i)10-s + (−0.477 − 0.275i)11-s + (−0.277 + 0.415i)12-s + (0.381 + 0.381i)13-s + (0.345 − 0.616i)14-s + (0.988 − 0.153i)15-s + (0.124 + 0.216i)16-s + (−1.45 + 0.389i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.947 + 0.318i$
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: \((2,\ 210,\ (\ :1),\ -0.947 + 0.318i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00971659 - 0.0594128i\)
\(L(\frac12)\) \(\approx\) \(0.00971659 - 0.0594128i\)
\(L(2)\) 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 \( 1 + (0.197 - 2.99i)T \)
5 \( 1 + (1.09 + 4.87i)T \)
7 \( 1 + (1.72 - 6.78i)T \)
good11 \( 1 + (5.25 + 3.03i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-4.95 - 4.95i)T + 169iT^{2} \)
17 \( 1 + (24.7 - 6.62i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (8.47 + 14.6i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-7.44 + 27.7i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + 29.7T + 841T^{2} \)
31 \( 1 + (36.5 + 21.1i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (12.4 - 46.2i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 2.99T + 1.68e3T^{2} \)
43 \( 1 + (-22.2 + 22.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (16.3 - 61.1i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (1.70 - 0.456i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-72.6 - 41.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (18.0 - 10.4i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-32.1 + 8.61i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 10.6iT - 5.04e3T^{2} \)
73 \( 1 + (-94.5 + 25.3i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-74.6 + 43.1i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (47.0 - 47.0i)T - 6.88e3iT^{2} \)
89 \( 1 + (138. - 80.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (81.7 - 81.7i)T - 9.40e3iT^{2} \)
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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

−12.52632669617282177840557358929, −11.38786944280377795096931350453, −10.81797199680446124180568871547, −9.430151635924837410845283943953, −8.910054847206432986067461228900, −8.254539255817303743064403059930, −6.42937141542441225323711420939, −5.24093977347330356296691557033, −4.07101281171003706966013796476, −2.41852769186474468951738999631, 0.03914069387645850111688845592, 2.00546324702959237605683959173, 3.55589140883506998852788115178, 5.69757765811103886006953705974, 6.93708233797779150745403239298, 7.29778268614692870750522595598, 8.340544476498451296181340424485, 9.663284486177212965352667302725, 10.93462423093275825104411115058, 11.14362597461667305669832767101

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