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Properties

Label 2-210-105.59-c1-0-4
Degree $2$
Conductor $210$
Sign $0.984 + 0.176i$
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 + (−1.72 − 0.178i)3-s + (−0.499 + 0.866i)4-s + (2.20 + 0.358i)5-s + (0.707 + 1.58i)6-s + (−1.41 + 2.23i)7-s + 0.999·8-s + (2.93 + 0.614i)9-s + (−0.792 − 2.09i)10-s + (0.184 + 0.106i)11-s + (1.01 − 1.40i)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 + (−0.994 − 0.102i)3-s + (−0.249 + 0.433i)4-s + (0.987 + 0.160i)5-s + (0.288 + 0.645i)6-s + (−0.534 + 0.845i)7-s + 0.353·8-s + (0.978 + 0.204i)9-s + (−0.250 − 0.661i)10-s + (0.0557 + 0.0321i)11-s + (0.293 − 0.404i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.176i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.984 + 0.176i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.984 + 0.176i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.866510 - 0.0772268i\)
\(L(\frac12)\) \(\approx\) \(0.866510 - 0.0772268i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 + 0.178i)T \)
5 \( 1 + (-2.20 - 0.358i)T \)
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

−12.13345996371691381079020077540, −11.30950245828017007659373516823, −10.55247275440808813393663719079, −9.478334369101174212603128590784, −8.804885729411226306453587481176, −7.02621583332069122309417964996, −6.06078602239631919122382696872, −5.13143956615204619931651802965, −3.25481151172314200032636796302, −1.51991327009372258440640246719, 1.17597364674788496620234932764, 3.94612139210766564989398273095, 5.36529187534257736350517956506, 6.30166169116995757080679101341, 6.91013826885834210956228736772, 8.491228175606489565991258516336, 9.555355723519345290838797884420, 10.47487600740412418289309796436, 11.03291003995246650464799786882, 12.57954516795498922030960971493

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