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Properties

Label 2-210-15.14-c2-0-20
Degree $2$
Conductor $210$
Sign $0.809 + 0.586i$
Analytic cond. $5.72208$
Root an. cond. $2.39208$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + (1.70 − 2.46i)3-s + 2.00·4-s + (4.99 + 0.104i)5-s + (2.41 − 3.48i)6-s + 2.64i·7-s + 2.82·8-s + (−3.15 − 8.42i)9-s + (7.06 + 0.148i)10-s + 2.66i·11-s + (3.41 − 4.93i)12-s + 5.56i·13-s + 3.74i·14-s + (8.80 − 12.1i)15-s + 4.00·16-s − 18.5·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.569 − 0.821i)3-s + 0.500·4-s + (0.999 + 0.0209i)5-s + (0.402 − 0.581i)6-s + 0.377i·7-s + 0.353·8-s + (−0.350 − 0.936i)9-s + (0.706 + 0.0148i)10-s + 0.242i·11-s + (0.284 − 0.410i)12-s + 0.427i·13-s + 0.267i·14-s + (0.586 − 0.809i)15-s + 0.250·16-s − 1.09·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 + 0.586i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.809 + 0.586i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.809 + 0.586i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.89819 - 0.939721i\)
\(L(\frac12)\) \(\approx\) \(2.89819 - 0.939721i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 + (-1.70 + 2.46i)T \)
5 \( 1 + (-4.99 - 0.104i)T \)
7 \( 1 - 2.64iT \)
good11 \( 1 - 2.66iT - 121T^{2} \)
13 \( 1 - 5.56iT - 169T^{2} \)
17 \( 1 + 18.5T + 289T^{2} \)
19 \( 1 - 0.634T + 361T^{2} \)
23 \( 1 + 15.6T + 529T^{2} \)
29 \( 1 + 43.3iT - 841T^{2} \)
31 \( 1 - 13.5T + 961T^{2} \)
37 \( 1 - 35.0iT - 1.36e3T^{2} \)
41 \( 1 - 15.6iT - 1.68e3T^{2} \)
43 \( 1 - 64.4iT - 1.84e3T^{2} \)
47 \( 1 + 52.5T + 2.20e3T^{2} \)
53 \( 1 - 51.6T + 2.80e3T^{2} \)
59 \( 1 - 32.9iT - 3.48e3T^{2} \)
61 \( 1 + 104.T + 3.72e3T^{2} \)
67 \( 1 - 113. iT - 4.48e3T^{2} \)
71 \( 1 + 36.8iT - 5.04e3T^{2} \)
73 \( 1 + 144. iT - 5.32e3T^{2} \)
79 \( 1 - 57.9T + 6.24e3T^{2} \)
83 \( 1 - 45.0T + 6.88e3T^{2} \)
89 \( 1 + 80.9iT - 7.92e3T^{2} \)
97 \( 1 - 96.2iT - 9.40e3T^{2} \)
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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

−12.24291045471510257255028659947, −11.42605253719473061997885487606, −9.994101035071670571905900017592, −9.058726291214366957756434647311, −7.925663023632646561075582402127, −6.60857855671042836739456537297, −6.04007079321087763760304286784, −4.52152661391382002758301562367, −2.79788645288258430191196257158, −1.79834760599680584362571055551, 2.15966841423324145990388753124, 3.47566119577092800481677241823, 4.73819267313942855745295277447, 5.71629419923919543188352628622, 6.97702836348241846382116861710, 8.403996790631790903077143803407, 9.383423099894247291495078764043, 10.42955929067225683887756395679, 10.99187367668518143884660189805, 12.48816439837346003748833471791

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