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Properties

Label 2-210-105.17-c2-0-16
Degree $2$
Conductor $210$
Sign $0.732 + 0.680i$
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 + (−0.897 + 2.86i)3-s + (1.73 + i)4-s + (4.28 − 2.57i)5-s + (2.27 − 3.58i)6-s + (−1.16 − 6.90i)7-s + (−1.99 − 2i)8-s + (−7.39 − 5.13i)9-s + (−6.79 + 1.94i)10-s + (7.83 + 4.52i)11-s + (−4.41 + 4.06i)12-s + (−15.3 − 15.3i)13-s + (−0.941 + 9.85i)14-s + (3.52 + 14.5i)15-s + (1.99 + 3.46i)16-s + (25.1 − 6.74i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.299 + 0.954i)3-s + (0.433 + 0.250i)4-s + (0.857 − 0.515i)5-s + (0.378 − 0.597i)6-s + (−0.165 − 0.986i)7-s + (−0.249 − 0.250i)8-s + (−0.821 − 0.570i)9-s + (−0.679 + 0.194i)10-s + (0.712 + 0.411i)11-s + (−0.368 + 0.338i)12-s + (−1.18 − 1.18i)13-s + (−0.0672 + 0.703i)14-s + (0.235 + 0.971i)15-s + (0.124 + 0.216i)16-s + (1.48 − 0.396i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.732 + 0.680i$
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: \((2,\ 210,\ (\ :1),\ 0.732 + 0.680i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.990987 - 0.389094i\)
\(L(\frac12)\) \(\approx\) \(0.990987 - 0.389094i\)
\(L(2)\) 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 \( 1 + (0.897 - 2.86i)T \)
5 \( 1 + (-4.28 + 2.57i)T \)
7 \( 1 + (1.16 + 6.90i)T \)
good11 \( 1 + (-7.83 - 4.52i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (15.3 + 15.3i)T + 169iT^{2} \)
17 \( 1 + (-25.1 + 6.74i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (0.500 + 0.867i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-3.42 + 12.7i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 8.69T + 841T^{2} \)
31 \( 1 + (-29.5 - 17.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-4.54 + 16.9i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 18.4T + 1.68e3T^{2} \)
43 \( 1 + (-24.0 + 24.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (8.15 - 30.4i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (47.9 - 12.8i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-78.7 - 45.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-56.9 + 32.8i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (13.1 - 3.51i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 86.5iT - 5.04e3T^{2} \)
73 \( 1 + (100. - 26.9i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (69.3 - 40.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (11.6 - 11.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (62.2 - 35.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-5.41 + 5.41i)T - 9.40e3iT^{2} \)
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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

−11.97114516495187320061303324482, −10.58184714937086980416625361487, −10.00172451624602753454819819269, −9.503541217718261288996788510648, −8.237478505743995122637017881612, −6.98023510752112952032427748883, −5.64157684596002900322645628516, −4.53762122280792352371119960566, −2.99535703343904819152683606716, −0.827731310075912502868458192301, 1.58419503494072618143890035385, 2.76743673258369497785015480293, 5.40392272323800000795661955986, 6.26685792661393872370165163712, 7.04579088985218756166315900423, 8.242686474727124451357364276904, 9.335527160941838977042907924124, 10.07964038695629520881238715860, 11.55886256317713634574850145307, 11.95709278695751476492746598028

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