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Properties

Label 2-210-105.89-c1-0-1
Degree $2$
Conductor $210$
Sign $-0.939 - 0.343i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−1.01 + 1.40i)3-s + (−0.499 − 0.866i)4-s + (0.792 + 2.09i)5-s + (−0.707 − 1.58i)6-s + (1.41 + 2.23i)7-s + 0.999·8-s + (−0.936 − 2.85i)9-s + (−2.20 − 0.358i)10-s + (−0.184 + 0.106i)11-s + (1.72 + 0.178i)12-s − 6.70·13-s + (−2.64 + 0.106i)14-s + (−3.73 − 1.01i)15-s + (−0.5 + 0.866i)16-s + (−2.73 + 1.58i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.586 + 0.809i)3-s + (−0.249 − 0.433i)4-s + (0.354 + 0.935i)5-s + (−0.288 − 0.645i)6-s + (0.534 + 0.845i)7-s + 0.353·8-s + (−0.312 − 0.950i)9-s + (−0.697 − 0.113i)10-s + (−0.0557 + 0.0321i)11-s + (0.497 + 0.0514i)12-s − 1.85·13-s + (−0.706 + 0.0285i)14-s + (−0.965 − 0.261i)15-s + (−0.125 + 0.216i)16-s + (−0.664 + 0.383i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.939 - 0.343i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ -0.939 - 0.343i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.132868 + 0.750190i\)
\(L(\frac12)\) \(\approx\) \(0.132868 + 0.750190i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (1.01 - 1.40i)T \)
5 \( 1 + (-0.792 - 2.09i)T \)
7 \( 1 + (-1.41 - 2.23i)T \)
good11 \( 1 + (0.184 - 0.106i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.70T + 13T^{2} \)
17 \( 1 + (2.73 - 1.58i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.23 - 2.44i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.23 + 5.60i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.02iT - 29T^{2} \)
31 \( 1 + (0.261 - 0.150i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.17 - 3.56i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.70T + 41T^{2} \)
43 \( 1 - 2.02iT - 43T^{2} \)
47 \( 1 + (-6.71 - 3.87i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.5 + 4.33i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.45 - 4.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 - 1.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.61 + 2.66i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.02iT - 71T^{2} \)
73 \( 1 + (5.99 + 10.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.261 - 0.452i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.7iT - 83T^{2} \)
89 \( 1 + (-5.28 + 9.15i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.50T + 97T^{2} \)
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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

−12.61275113316656998123087716871, −11.61827877116912419843106997812, −10.69640434687997577899893456509, −9.832945359692112540256653657639, −9.078427509412455023908190246813, −7.68744235981889151790107004942, −6.56848612863629966389049215409, −5.55894043339607881073954013777, −4.61914515318260965349340190319, −2.64887233708006081819641902319, 0.796904557158033602697725363193, 2.30666811375094055621543682092, 4.56475277346303233806449854941, 5.38577905637631234615851977469, 7.19319260314978433528497091331, 7.73255128212587238932359803283, 9.130920885858556980145160025492, 10.00387115828109036534325937561, 11.22010396106608420342163470424, 11.85052056754111344300009161835

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