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Properties

Label 2-210-105.38-c2-0-0
Degree $2$
Conductor $210$
Sign $-0.950 - 0.309i$
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  + (−0.366 + 1.36i)2-s + (−1.66 − 2.49i)3-s + (−1.73 − i)4-s + (1.33 − 4.81i)5-s + (4.01 − 1.35i)6-s + (−6.98 + 0.400i)7-s + (2 − 1.99i)8-s + (−3.46 + 8.30i)9-s + (6.08 + 3.59i)10-s + (4.21 + 2.43i)11-s + (0.386 + 5.98i)12-s + (−13.3 + 13.3i)13-s + (2.01 − 9.69i)14-s + (−14.2 + 4.67i)15-s + (1.99 + 3.46i)16-s + (6.77 + 25.3i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.554 − 0.832i)3-s + (−0.433 − 0.250i)4-s + (0.267 − 0.963i)5-s + (0.669 − 0.226i)6-s + (−0.998 + 0.0572i)7-s + (0.250 − 0.249i)8-s + (−0.384 + 0.923i)9-s + (0.608 + 0.359i)10-s + (0.382 + 0.221i)11-s + (0.0321 + 0.498i)12-s + (−1.02 + 1.02i)13-s + (0.143 − 0.692i)14-s + (−0.950 + 0.311i)15-s + (0.124 + 0.216i)16-s + (0.398 + 1.48i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.309i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.950 - 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.950 - 0.309i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.950 - 0.309i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0133854 + 0.0843813i\)
\(L(\frac12)\) \(\approx\) \(0.0133854 + 0.0843813i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 - 1.36i)T \)
3 \( 1 + (1.66 + 2.49i)T \)
5 \( 1 + (-1.33 + 4.81i)T \)
7 \( 1 + (6.98 - 0.400i)T \)
good11 \( 1 + (-4.21 - 2.43i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (13.3 - 13.3i)T - 169iT^{2} \)
17 \( 1 + (-6.77 - 25.3i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (6.44 + 11.1i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (18.8 + 5.04i)T + (458. + 264.5i)T^{2} \)
29 \( 1 + 41.0T + 841T^{2} \)
31 \( 1 + (-28.4 - 16.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (60.2 + 16.1i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 - 50.2T + 1.68e3T^{2} \)
43 \( 1 + (39.5 + 39.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (-3.46 - 0.928i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-14.9 - 55.8i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (41.0 + 23.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (58.7 - 33.9i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (10.7 + 40.3i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 30.7iT - 5.04e3T^{2} \)
73 \( 1 + (-14.0 - 52.4i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-26.6 + 15.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (43.6 + 43.6i)T + 6.88e3iT^{2} \)
89 \( 1 + (-54.3 + 31.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (16.7 + 16.7i)T + 9.40e3iT^{2} \)
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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

−12.49403813134816391077135105081, −12.09786936673909702216975307632, −10.47303712764598928468191098326, −9.431607078018705611349972996292, −8.559631884324628753832835480108, −7.34529573605918908350426543885, −6.43226454166302117342295726201, −5.58364274838241550596249975562, −4.28615605888801253160273557255, −1.82026458768013047608846891342, 0.05204335103970043642844405001, 2.80774106612352627343799317442, 3.69994116584969589694366017067, 5.29212296145517484720238844606, 6.37471378397095179141601423637, 7.63152133209424688007089707866, 9.374251081027933167639751279202, 9.891442343652962940644108260256, 10.55046309502590625523298904212, 11.61610620161146790913045561233

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