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Properties

Label 2-210-21.2-c2-0-13
Degree $2$
Conductor $210$
Sign $0.999 - 0.0293i$
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 + (−2.95 − 0.513i)3-s + (0.999 + 1.73i)4-s + (−1.93 − 1.11i)5-s + (−3.25 − 2.71i)6-s + (5.52 − 4.30i)7-s + 2.82i·8-s + (8.47 + 3.03i)9-s + (−1.58 − 2.73i)10-s + (1.44 − 0.836i)11-s + (−2.06 − 5.63i)12-s + 20.4·13-s + (9.80 − 1.36i)14-s + (5.14 + 4.29i)15-s + (−2.00 + 3.46i)16-s + (3.30 − 1.90i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.985 − 0.171i)3-s + (0.249 + 0.433i)4-s + (−0.387 − 0.223i)5-s + (−0.542 − 0.453i)6-s + (0.788 − 0.614i)7-s + 0.353i·8-s + (0.941 + 0.337i)9-s + (−0.158 − 0.273i)10-s + (0.131 − 0.0760i)11-s + (−0.172 − 0.469i)12-s + 1.57·13-s + (0.700 − 0.0973i)14-s + (0.343 + 0.286i)15-s + (−0.125 + 0.216i)16-s + (0.194 − 0.112i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0293i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.999 - 0.0293i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.999 - 0.0293i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.73301 + 0.0254694i\)
\(L(\frac12)\) \(\approx\) \(1.73301 + 0.0254694i\)
\(L(2)\) 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 \( 1 + (2.95 + 0.513i)T \)
5 \( 1 + (1.93 + 1.11i)T \)
7 \( 1 + (-5.52 + 4.30i)T \)
good11 \( 1 + (-1.44 + 0.836i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 20.4T + 169T^{2} \)
17 \( 1 + (-3.30 + 1.90i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-8.08 + 14.0i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-26.8 - 15.4i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 15.5iT - 841T^{2} \)
31 \( 1 + (21.5 + 37.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (15.1 - 26.1i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 45.4iT - 1.68e3T^{2} \)
43 \( 1 - 33.1T + 1.84e3T^{2} \)
47 \( 1 + (18.0 + 10.4i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (29.2 - 16.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (67.5 - 38.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (44.2 - 76.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-17.6 - 30.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 111. iT - 5.04e3T^{2} \)
73 \( 1 + (-65.5 - 113. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-26.0 + 45.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 72.2iT - 6.88e3T^{2} \)
89 \( 1 + (-79.2 - 45.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 34.3T + 9.40e3T^{2} \)
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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

−12.07762118127583472797174303421, −11.20870760021881441920051844611, −10.78128415169412576471322905617, −9.043765839249623312863120735844, −7.76958785619313981670394464753, −6.96579882557853387413183287038, −5.76278944886156527303156965222, −4.80175728553653555800992966947, −3.71662089673243324567423716604, −1.19048262019729086424932830561, 1.40122318227317072529438308308, 3.48490676948511859639181825307, 4.70741753453464005812183426750, 5.71198541241103278902213954740, 6.65707533808814270700794372529, 8.070899948040176622277747391404, 9.358725165390976477888673282482, 10.89572934755296814743560824730, 11.00862535152598311231150813091, 12.11421778050677805884234876290

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