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Properties

Label 2-210-105.17-c2-0-26
Degree $2$
Conductor $210$
Sign $0.260 + 0.965i$
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  + (1.36 + 0.366i)2-s + (−0.489 − 2.95i)3-s + (1.73 + i)4-s + (1.47 − 4.77i)5-s + (0.414 − 4.22i)6-s + (6.99 + 0.132i)7-s + (1.99 + 2i)8-s + (−8.52 + 2.89i)9-s + (3.76 − 5.98i)10-s + (−9.30 − 5.36i)11-s + (2.11 − 5.61i)12-s + (−3.28 − 3.28i)13-s + (9.51 + 2.74i)14-s + (−14.8 − 2.02i)15-s + (1.99 + 3.46i)16-s + (16.0 − 4.29i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.163 − 0.986i)3-s + (0.433 + 0.250i)4-s + (0.295 − 0.955i)5-s + (0.0690 − 0.703i)6-s + (0.999 + 0.0189i)7-s + (0.249 + 0.250i)8-s + (−0.946 + 0.322i)9-s + (0.376 − 0.598i)10-s + (−0.845 − 0.488i)11-s + (0.175 − 0.468i)12-s + (−0.252 − 0.252i)13-s + (0.679 + 0.195i)14-s + (−0.990 − 0.135i)15-s + (0.124 + 0.216i)16-s + (0.942 − 0.252i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.260 + 0.965i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.260 + 0.965i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.82100 - 1.39408i\)
\(L(\frac12)\) \(\approx\) \(1.82100 - 1.39408i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
3 \( 1 + (0.489 + 2.95i)T \)
5 \( 1 + (-1.47 + 4.77i)T \)
7 \( 1 + (-6.99 - 0.132i)T \)
good11 \( 1 + (9.30 + 5.36i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (3.28 + 3.28i)T + 169iT^{2} \)
17 \( 1 + (-16.0 + 4.29i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (1.04 + 1.81i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-3.40 + 12.7i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 21.8T + 841T^{2} \)
31 \( 1 + (-40.3 - 23.3i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (10.9 - 40.7i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 41.7T + 1.68e3T^{2} \)
43 \( 1 + (32.4 - 32.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (18.0 - 67.3i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-31.2 + 8.36i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-8.21 - 4.74i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-100. + 58.3i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (2.25 - 0.604i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 82.6iT - 5.04e3T^{2} \)
73 \( 1 + (-89.0 + 23.8i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (27.5 - 15.9i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (37.0 - 37.0i)T - 6.88e3iT^{2} \)
89 \( 1 + (136. - 78.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (41.8 - 41.8i)T - 9.40e3iT^{2} \)
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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

−12.15439629425510186880159546641, −11.39710984864426421003220946220, −10.15329270067386080683167771186, −8.304944601707849683226379984110, −8.119459916441793925506512679669, −6.67814895503756689510299340016, −5.43376173939798706115496256078, −4.85556491041249701068410876999, −2.74842549538035282741374963022, −1.18602824150013568849265543121, 2.32686046283312902350514744660, 3.65409721961648063895864997016, 4.91681092422591734842534347562, 5.72690130798709246989504847387, 7.12796417843828324831413216228, 8.325740458570890390757541065260, 9.973861926530424739193071328050, 10.35197114630165588841353459787, 11.38903028369013057216295124732, 12.04507142944253064625679201349

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