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Properties

Label 2-210-105.17-c2-0-1
Degree $2$
Conductor $210$
Sign $0.548 - 0.835i$
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 + (−2.72 − 1.26i)3-s + (1.73 + i)4-s + (−2.34 − 4.41i)5-s + (3.25 + 2.72i)6-s + (−6.07 + 3.47i)7-s + (−1.99 − 2i)8-s + (5.80 + 6.87i)9-s + (1.57 + 6.89i)10-s + (−5.23 − 3.02i)11-s + (−3.45 − 4.90i)12-s + (−7.33 − 7.33i)13-s + (9.57 − 2.51i)14-s + (0.787 + 14.9i)15-s + (1.99 + 3.46i)16-s + (22.5 − 6.03i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.907 − 0.420i)3-s + (0.433 + 0.250i)4-s + (−0.468 − 0.883i)5-s + (0.542 + 0.453i)6-s + (−0.868 + 0.495i)7-s + (−0.249 − 0.250i)8-s + (0.645 + 0.763i)9-s + (0.157 + 0.689i)10-s + (−0.475 − 0.274i)11-s + (−0.287 − 0.409i)12-s + (−0.564 − 0.564i)13-s + (0.683 − 0.179i)14-s + (0.0524 + 0.998i)15-s + (0.124 + 0.216i)16-s + (1.32 − 0.354i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 - 0.835i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.548 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.548 - 0.835i$
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: \((2,\ 210,\ (\ :1),\ 0.548 - 0.835i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.333440 + 0.179949i\)
\(L(\frac12)\) \(\approx\) \(0.333440 + 0.179949i\)
\(L(2)\) 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 \( 1 + (2.72 + 1.26i)T \)
5 \( 1 + (2.34 + 4.41i)T \)
7 \( 1 + (6.07 - 3.47i)T \)
good11 \( 1 + (5.23 + 3.02i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (7.33 + 7.33i)T + 169iT^{2} \)
17 \( 1 + (-22.5 + 6.03i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-17.0 - 29.6i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (7.11 - 26.5i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + 6.86T + 841T^{2} \)
31 \( 1 + (-38.9 - 22.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (9.14 - 34.1i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 18.2T + 1.68e3T^{2} \)
43 \( 1 + (-17.2 + 17.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (6.24 - 23.3i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (77.2 - 20.7i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (97.9 + 56.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-16.8 + 9.74i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-10.9 + 2.92i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 80.6iT - 5.04e3T^{2} \)
73 \( 1 + (1.93 - 0.519i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (73.1 - 42.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (4.56 - 4.56i)T - 6.88e3iT^{2} \)
89 \( 1 + (21.5 - 12.4i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-38.4 + 38.4i)T - 9.40e3iT^{2} \)
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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

−12.26889213716255137192593676577, −11.54891746303423182781104084498, −10.15469757422878213955146958140, −9.608167759716497972384653401293, −8.084249956920833239181649359975, −7.53130533091062620527605148666, −5.99037660096448464004497856061, −5.19902753825210433941533706998, −3.27507140213886168517030335655, −1.18670673701333240532615232879, 0.34304388518142599043819954113, 2.98409468940195982357742329686, 4.48599012485209136349243417450, 6.01840436268102199890066250036, 6.93407254575876115336136866315, 7.66028979712512392785561377283, 9.407460441843226668981185054091, 10.09536191966194543394709285349, 10.81636896366446474875809510560, 11.75264766536534011802210249128

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