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Properties

Label 2-210-7.6-c2-0-0
Degree $2$
Conductor $210$
Sign $-0.957 + 0.288i$
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 + 1.73i·3-s + 2.00·4-s + 2.23i·5-s − 2.44i·6-s + (−6.70 + 2.02i)7-s − 2.82·8-s − 2.99·9-s − 3.16i·10-s − 5.00·11-s + 3.46i·12-s − 9.06i·13-s + (9.47 − 2.86i)14-s − 3.87·15-s + 4.00·16-s + 19.7i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577i·3-s + 0.500·4-s + 0.447i·5-s − 0.408i·6-s + (−0.957 + 0.288i)7-s − 0.353·8-s − 0.333·9-s − 0.316i·10-s − 0.455·11-s + 0.288i·12-s − 0.697i·13-s + (0.676 − 0.204i)14-s − 0.258·15-s + 0.250·16-s + 1.16i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.957 + 0.288i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.957 + 0.288i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0285923 - 0.193684i\)
\(L(\frac12)\) \(\approx\) \(0.0285923 - 0.193684i\)
\(L(2)\) 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 \( 1 - 1.73iT \)
5 \( 1 - 2.23iT \)
7 \( 1 + (6.70 - 2.02i)T \)
good11 \( 1 + 5.00T + 121T^{2} \)
13 \( 1 + 9.06iT - 169T^{2} \)
17 \( 1 - 19.7iT - 289T^{2} \)
19 \( 1 + 25.6iT - 361T^{2} \)
23 \( 1 + 40.9T + 529T^{2} \)
29 \( 1 + 22.3T + 841T^{2} \)
31 \( 1 + 15.9iT - 961T^{2} \)
37 \( 1 + 44.7T + 1.36e3T^{2} \)
41 \( 1 - 27.0iT - 1.68e3T^{2} \)
43 \( 1 + 30.6T + 1.84e3T^{2} \)
47 \( 1 - 58.1iT - 2.20e3T^{2} \)
53 \( 1 - 65.6T + 2.80e3T^{2} \)
59 \( 1 + 32.5iT - 3.48e3T^{2} \)
61 \( 1 - 83.5iT - 3.72e3T^{2} \)
67 \( 1 - 72.0T + 4.48e3T^{2} \)
71 \( 1 + 24.8T + 5.04e3T^{2} \)
73 \( 1 - 67.8iT - 5.32e3T^{2} \)
79 \( 1 - 30.4T + 6.24e3T^{2} \)
83 \( 1 - 72.4iT - 6.88e3T^{2} \)
89 \( 1 + 113. iT - 7.92e3T^{2} \)
97 \( 1 - 103. iT - 9.40e3T^{2} \)
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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

−12.57914539149010603216612122901, −11.43855214022334836499109678235, −10.43224968155618424721629027866, −9.899065920066564533147646317387, −8.838988910525826823008023210113, −7.82184864898721886492414522022, −6.54695282392107001394671217423, −5.58691772237443574103303752661, −3.76158279374081791971704362091, −2.51754508473643973126804886287, 0.12726194643307457236450208136, 1.96947072880861016113678569573, 3.65876682380825580706560970191, 5.53154024513644878697823257672, 6.66927837949969088599758732479, 7.58175549795877651929007416857, 8.610903406957405007410591894417, 9.644615279544272350571113881618, 10.39409146712502877505938206925, 11.81546914520498869738069075759

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