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Properties

Label 2-210-21.5-c1-0-0
Degree $2$
Conductor $210$
Sign $0.342 - 0.939i$
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  + (−0.866 − 0.5i)2-s + (−1.72 + 0.111i)3-s + (0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (1.55 + 0.767i)6-s + (−1.91 + 1.82i)7-s − 0.999i·8-s + (2.97 − 0.385i)9-s + (−0.866 + 0.499i)10-s + (−1.99 + 1.15i)11-s + (−0.960 − 1.44i)12-s + 5.00i·13-s + (2.57 − 0.618i)14-s + (−0.767 + 1.55i)15-s + (−0.5 + 0.866i)16-s + (3.15 + 5.45i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.997 + 0.0644i)3-s + (0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + (0.633 + 0.313i)6-s + (−0.725 + 0.688i)7-s − 0.353i·8-s + (0.991 − 0.128i)9-s + (−0.273 + 0.158i)10-s + (−0.602 + 0.347i)11-s + (−0.277 − 0.416i)12-s + 1.38i·13-s + (0.687 − 0.165i)14-s + (−0.198 + 0.400i)15-s + (−0.125 + 0.216i)16-s + (0.764 + 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.342 - 0.939i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.342 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.420672 + 0.294264i\)
\(L(\frac12)\) \(\approx\) \(0.420672 + 0.294264i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (1.72 - 0.111i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (1.91 - 1.82i)T \)
good11 \( 1 + (1.99 - 1.15i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.00iT - 13T^{2} \)
17 \( 1 + (-3.15 - 5.45i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.54 - 3.77i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.51 + 3.18i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.83iT - 29T^{2} \)
31 \( 1 + (3.79 - 2.19i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.45 + 5.98i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.79T + 41T^{2} \)
43 \( 1 - 2.55T + 43T^{2} \)
47 \( 1 + (-0.828 + 1.43i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.81 + 1.62i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.96 - 8.60i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.18 + 2.99i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.38 - 2.39i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.85iT - 71T^{2} \)
73 \( 1 + (3.18 - 1.84i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.27 - 2.21i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.83T + 83T^{2} \)
89 \( 1 + (-2.94 + 5.09i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.61iT - 97T^{2} \)
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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

−12.25577560818224522040122989799, −11.77942189694847149187001859134, −10.40741700062802874629626078419, −9.844252863567403023531589041737, −8.847681564599172401616223148267, −7.51962881539441978454273903953, −6.30994368983703329243577196749, −5.38633088802435002833285239887, −3.81366168170160508571584095722, −1.77306814410922867735843027912, 0.61171406472908033165882173892, 3.15557461016795787422721657125, 5.17926024203838608395075728703, 5.97680959983451333741678766410, 7.20994812264486664639040961444, 7.77278920124397683423623921054, 9.729123127788617429300486427911, 10.02132617148095414083565647839, 11.07899245883078003362122414776, 11.93590381765889276862253569007

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