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Properties

Label 2-210-105.89-c1-0-5
Degree $2$
Conductor $210$
Sign $0.515 - 0.857i$
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 + (0.686 − 2.12i)5-s + (−1.68 + 0.396i)6-s + (2.5 + 0.866i)7-s + 0.999·8-s + (−0.186 + 2.99i)9-s + (1.5 + 1.65i)10-s + (0.813 − 0.469i)11-s + (0.500 − 1.65i)12-s + 2·13-s + (−2 + 1.73i)14-s + (3.5 − 1.65i)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 + (0.306 − 0.951i)5-s + (−0.688 + 0.161i)6-s + (0.944 + 0.327i)7-s + 0.353·8-s + (−0.0620 + 0.998i)9-s + (0.474 + 0.524i)10-s + (0.245 − 0.141i)11-s + (0.144 − 0.478i)12-s + 0.554·13-s + (−0.534 + 0.462i)14-s + (0.903 − 0.428i)15-s + (−0.125 + 0.216i)16-s + (−1.39 + 0.804i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.515 - 0.857i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.515 - 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17663 + 0.665478i\)
\(L(\frac12)\) \(\approx\) \(1.17663 + 0.665478i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.686 + 2.12i)T \)
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

−12.78665339993164875945038740876, −11.27914470851601814860616866362, −10.44443307054541295888751043600, −9.101298713781177224477914644211, −8.714165136287436719251607465676, −7.946114334651656511953912994291, −6.27703381296364396863576202218, −5.01005213489748806786828141599, −4.20781049770100152978706835380, −1.95513667344711490004689839021, 1.69202001641479529898607694123, 2.88566775485323823750553858300, 4.32708014018596054336369281595, 6.34861727149625616978157756234, 7.28479412697082408594346692964, 8.281692311837272084729167176943, 9.187680073653195972287960208497, 10.37919991182282742146317803349, 11.24816106678372933623686090720, 12.02920218061278566476916354980

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