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Properties

Label 2-210-21.11-c2-0-9
Degree $2$
Conductor $210$
Sign $0.393 + 0.919i$
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  + (−1.22 + 0.707i)2-s + (1.03 − 2.81i)3-s + (0.999 − 1.73i)4-s + (1.93 − 1.11i)5-s + (0.716 + 4.18i)6-s + (6.69 − 2.05i)7-s + 2.82i·8-s + (−6.83 − 5.85i)9-s + (−1.58 + 2.73i)10-s + (5.25 + 3.03i)11-s + (−3.83 − 4.61i)12-s + 4.30·13-s + (−6.74 + 7.24i)14-s + (−1.13 − 6.61i)15-s + (−2.00 − 3.46i)16-s + (−11.9 − 6.89i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.346 − 0.938i)3-s + (0.249 − 0.433i)4-s + (0.387 − 0.223i)5-s + (0.119 + 0.696i)6-s + (0.956 − 0.293i)7-s + 0.353i·8-s + (−0.759 − 0.650i)9-s + (−0.158 + 0.273i)10-s + (0.477 + 0.275i)11-s + (−0.319 − 0.384i)12-s + 0.331·13-s + (−0.481 + 0.517i)14-s + (−0.0755 − 0.440i)15-s + (−0.125 − 0.216i)16-s + (−0.702 − 0.405i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.393 + 0.919i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.393 + 0.919i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.393 + 0.919i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.20460 - 0.795144i\)
\(L(\frac12)\) \(\approx\) \(1.20460 - 0.795144i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
3 \( 1 + (-1.03 + 2.81i)T \)
5 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 + (-6.69 + 2.05i)T \)
good11 \( 1 + (-5.25 - 3.03i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 4.30T + 169T^{2} \)
17 \( 1 + (11.9 + 6.89i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (5.01 + 8.68i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-12.6 + 7.27i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 50.2iT - 841T^{2} \)
31 \( 1 + (-4.34 + 7.52i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-30.8 - 53.4i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 5.51iT - 1.68e3T^{2} \)
43 \( 1 - 7.89T + 1.84e3T^{2} \)
47 \( 1 + (19.8 - 11.4i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-23.1 - 13.3i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (75.3 + 43.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-2.68 - 4.64i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (59.4 - 102. i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 22.8iT - 5.04e3T^{2} \)
73 \( 1 + (40.7 - 70.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-49.9 - 86.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 141. iT - 6.88e3T^{2} \)
89 \( 1 + (-76.8 + 44.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 130.T + 9.40e3T^{2} \)
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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

−11.79509435776211163020566505990, −11.12791292345971996201310600715, −9.737504435994580423907724307608, −8.759409373552721035900458965893, −8.005212037068551834106696748060, −6.98248446816628497644128909662, −6.06337236818198878306530265120, −4.56124253038281070154286437367, −2.36775430364901230892870515720, −1.04782150790736390116786268265, 1.87075686210135350374247560290, 3.37238568253205004997514475296, 4.70303437734158427875406505375, 6.03278204156455724014292436281, 7.60434947104674212679526775703, 8.777963490754259867387506650056, 9.164676898892973063721442075997, 10.59644704256448053062282828846, 10.91718667499999324479352560931, 11.99707546820983081076079445845

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