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Properties

Label 2-210-35.9-c1-0-3
Degree $2$
Conductor $210$
Sign $0.696 - 0.717i$
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 + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.806 + 2.08i)5-s + 0.999·6-s + (1.45 + 2.20i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−1.74 + 1.40i)10-s + (−3.05 − 5.28i)11-s + (0.866 + 0.499i)12-s + 1.68i·13-s + (0.155 + 2.64i)14-s + (0.344 + 2.20i)15-s + (−0.5 + 0.866i)16-s + (5.92 − 3.41i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.360 + 0.932i)5-s + 0.408·6-s + (0.550 + 0.835i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.550 + 0.443i)10-s + (−0.920 − 1.59i)11-s + (0.249 + 0.144i)12-s + 0.468i·13-s + (0.0416 + 0.705i)14-s + (0.0888 + 0.570i)15-s + (−0.125 + 0.216i)16-s + (1.43 − 0.829i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.696 - 0.717i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.696 - 0.717i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69680 + 0.717839i\)
\(L(\frac12)\) \(\approx\) \(1.69680 + 0.717839i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (0.806 - 2.08i)T \)
7 \( 1 + (-1.45 - 2.20i)T \)
good11 \( 1 + (3.05 + 5.28i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.68iT - 13T^{2} \)
17 \( 1 + (-5.92 + 3.41i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.844 + 1.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.00 + 1.15i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.41T + 29T^{2} \)
31 \( 1 + (-0.344 - 0.596i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.00 - 1.15i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.14T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + (2.65 + 1.53i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.05 - 5.23i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.52 + 6.09i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.73 - 8.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.5 + 6.10i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-3.46 + 2i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.65 + 6.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.5iT - 83T^{2} \)
89 \( 1 + (6.10 - 10.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.95iT - 97T^{2} \)
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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

−12.53872535330270988902556009302, −11.56778976704630896077102446491, −10.91510872218043401794671624925, −9.385048485540184108423484197760, −8.116910139720859853068991679748, −7.59351768993008019156504100778, −6.22071226175592948268364552045, −5.28019778152727577956884506333, −3.50593206615089233627659328247, −2.58878454601994060807560626861, 1.72127552125425944477238926834, 3.63925073547285117400037583568, 4.59815130006060255732935901431, 5.51343154845718137936800403794, 7.57822907955128719403641538481, 7.940725339225455815116681709343, 9.610330786292652655014273352218, 10.23051861832314827265084531397, 11.34972902441032215009289330551, 12.67415793430104084585058072982

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