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Properties

Label 2-210-35.12-c1-0-2
Degree $2$
Conductor $210$
Sign $0.768 - 0.640i$
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  + (0.965 − 0.258i)2-s + (−0.258 + 0.965i)3-s + (0.866 − 0.499i)4-s + (1.20 + 1.88i)5-s + i·6-s + (−0.781 + 2.52i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (1.64 + 1.50i)10-s + (−1.31 − 2.27i)11-s + (0.258 + 0.965i)12-s + (−1.21 − 1.21i)13-s + (−0.101 + 2.64i)14-s + (−2.13 + 0.674i)15-s + (0.500 − 0.866i)16-s + (7.31 + 1.95i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.149 + 0.557i)3-s + (0.433 − 0.249i)4-s + (0.538 + 0.842i)5-s + 0.408i·6-s + (−0.295 + 0.955i)7-s + (0.249 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.521 + 0.477i)10-s + (−0.395 − 0.685i)11-s + (0.0747 + 0.278i)12-s + (−0.337 − 0.337i)13-s + (−0.0270 + 0.706i)14-s + (−0.550 + 0.174i)15-s + (0.125 − 0.216i)16-s + (1.77 + 0.475i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.768 - 0.640i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.768 - 0.640i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62513 + 0.588526i\)
\(L(\frac12)\) \(\approx\) \(1.62513 + 0.588526i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (-1.20 - 1.88i)T \)
7 \( 1 + (0.781 - 2.52i)T \)
good11 \( 1 + (1.31 + 2.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.21 + 1.21i)T + 13iT^{2} \)
17 \( 1 + (-7.31 - 1.95i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.32 + 4.02i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.32 + 4.95i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 5.99iT - 29T^{2} \)
31 \( 1 + (8.66 - 5.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.82 - 1.02i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 5.59iT - 41T^{2} \)
43 \( 1 + (0.545 - 0.545i)T - 43iT^{2} \)
47 \( 1 + (1.64 + 6.12i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-8.28 - 2.22i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.86 - 6.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.16 - 2.40i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.663 - 2.47i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.36T + 71T^{2} \)
73 \( 1 + (3.53 - 13.1i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.78 + 4.49i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.99 - 7.99i)T + 83iT^{2} \)
89 \( 1 + (-0.0812 + 0.140i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.35 + 4.35i)T - 97iT^{2} \)
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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

−12.41424911523418404236866020077, −11.56822186493059096908051642917, −10.50314405758628496827823953419, −9.906852769885009365951431252764, −8.646693865733371243762955232680, −7.15742210606379012268368238089, −5.82006088578409514784670621023, −5.37960716191249333447522771184, −3.51101892217639977268113579691, −2.57463971265045979961032678764, 1.58124616217310837401481625619, 3.54055566082208968445205624270, 5.01056794996128995838115353881, 5.82712809529990988578838499656, 7.24855283899430920796927181669, 7.81024826961156756539043600349, 9.476694000744558209468452958188, 10.25290178703083654084942678471, 11.68755435954248172954832514302, 12.50588203253216899031238903667

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