Properties

Label 2-210-35.24-c2-0-4
Degree $2$
Conductor $210$
Sign $0.942 - 0.334i$
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 + (0.866 − 1.5i)3-s + (0.999 − 1.73i)4-s + (4.96 + 0.613i)5-s + 2.44i·6-s + (2.31 + 6.60i)7-s + 2.82i·8-s + (−1.5 − 2.59i)9-s + (−6.51 + 2.75i)10-s + (−0.676 + 1.17i)11-s + (−1.73 − 2.99i)12-s + 5.90·13-s + (−7.50 − 6.45i)14-s + (5.21 − 6.91i)15-s + (−2.00 − 3.46i)16-s + (−6.21 + 10.7i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.288 − 0.5i)3-s + (0.249 − 0.433i)4-s + (0.992 + 0.122i)5-s + 0.408i·6-s + (0.330 + 0.943i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.651 + 0.275i)10-s + (−0.0615 + 0.106i)11-s + (−0.144 − 0.249i)12-s + 0.454·13-s + (−0.536 − 0.461i)14-s + (0.347 − 0.460i)15-s + (−0.125 − 0.216i)16-s + (−0.365 + 0.633i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.942 - 0.334i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.942 - 0.334i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.53203 + 0.263617i\)
\(L(\frac12)\) \(\approx\) \(1.53203 + 0.263617i\)
\(L(2)\) 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 \( 1 + (-0.866 + 1.5i)T \)
5 \( 1 + (-4.96 - 0.613i)T \)
7 \( 1 + (-2.31 - 6.60i)T \)
good11 \( 1 + (0.676 - 1.17i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 5.90T + 169T^{2} \)
17 \( 1 + (6.21 - 10.7i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-27.5 + 15.9i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-1.31 + 0.758i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 15.0T + 841T^{2} \)
31 \( 1 + (-40.3 - 23.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-0.441 + 0.254i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 36.0iT - 1.68e3T^{2} \)
43 \( 1 + 67.6iT - 1.84e3T^{2} \)
47 \( 1 + (24.7 + 42.9i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-45.5 - 26.3i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (62.0 + 35.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-50.6 + 29.2i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (109. + 63.1i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 55.4T + 5.04e3T^{2} \)
73 \( 1 + (26.5 - 46.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-19.9 - 34.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 120.T + 6.88e3T^{2} \)
89 \( 1 + (5.68 - 3.28i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 154.T + 9.40e3T^{2} \)
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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

−12.15871066323140390316380403531, −11.16009668097245827181630913350, −9.998337951510706759947047499555, −9.069424734013721710826599525227, −8.383456550999502387240535127383, −7.10365125726318027725068213652, −6.12034864634700562030394064536, −5.16725199390138838369918940910, −2.79532586641662203542238241537, −1.52474648721794828808924898937, 1.29808458801679073648646143207, 2.96612606625746647624005487444, 4.41539848977925273723111918790, 5.80854219033667707786806653719, 7.22813331913744336510710343742, 8.265449108753828967230018735184, 9.412996675132943540710752167472, 9.999558137466791173537596045453, 10.87585443791206237541917346828, 11.81228135115170282532970694975

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