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Properties

Label 2-210-105.59-c1-0-0
Degree $2$
Conductor $210$
Sign $-0.821 - 0.569i$
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 + (−1.18 + 1.26i)3-s + (−0.499 + 0.866i)4-s + (1.5 + 1.65i)5-s + (−1.68 − 0.396i)6-s + (−2.5 + 0.866i)7-s − 0.999·8-s + (−0.186 − 2.99i)9-s + (−0.686 + 2.12i)10-s + (0.813 + 0.469i)11-s + (−0.500 − 1.65i)12-s − 2·13-s + (−2 − 1.73i)14-s + (−3.87 − 0.0737i)15-s + (−0.5 − 0.866i)16-s + (5.74 + 3.31i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.684 + 0.728i)3-s + (−0.249 + 0.433i)4-s + (0.670 + 0.741i)5-s + (−0.688 − 0.161i)6-s + (−0.944 + 0.327i)7-s − 0.353·8-s + (−0.0620 − 0.998i)9-s + (−0.216 + 0.672i)10-s + (0.245 + 0.141i)11-s + (−0.144 − 0.478i)12-s − 0.554·13-s + (−0.534 − 0.462i)14-s + (−0.999 − 0.0190i)15-s + (−0.125 − 0.216i)16-s + (1.39 + 0.804i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.821 - 0.569i$
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: \((2,\ 210,\ (\ :1/2),\ -0.821 - 0.569i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.319579 + 1.02221i\)
\(L(\frac12)\) \(\approx\) \(0.319579 + 1.02221i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.18 - 1.26i)T \)
5 \( 1 + (-1.5 - 1.65i)T \)
7 \( 1 + (2.5 - 0.866i)T \)
good11 \( 1 + (-0.813 - 0.469i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (-5.74 - 3.31i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.686 + 1.18i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.31iT - 29T^{2} \)
31 \( 1 + (-6.55 - 3.78i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.11 + 4.10i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.37T + 41T^{2} \)
43 \( 1 + 1.08iT - 43T^{2} \)
47 \( 1 + (-7.37 + 4.25i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.18 - 3.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.55 - 11.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.0 - 6.38i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.05 + 1.18i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.51iT - 71T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.55 + 7.89i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.8iT - 83T^{2} \)
89 \( 1 + (0.686 + 1.18i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.1T + 97T^{2} \)
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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

−12.61795935744981002616865455632, −12.02887913963393400268271948799, −10.55639371946922052186029091092, −9.956130264270265554536980611346, −9.031961126467121765048246567950, −7.41956053095984891250775878257, −6.12444944575195339898110914673, −5.88868914188935807420632921639, −4.28538566094244230463954409208, −2.99766768252635341737852230372, 0.937759310051651592901227003732, 2.68727074443064898284934803942, 4.55532473929117568496186694290, 5.67408968188284297306057447374, 6.53848819121091112833528499951, 7.86508457166676044497438649880, 9.394499244420321066657714326897, 10.03750497878771617823655569701, 11.20029487959412788916310963184, 12.29710024577713072487104038690

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