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Properties

Label 2-210-21.11-c2-0-15
Degree $2$
Conductor $210$
Sign $-0.639 + 0.768i$
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 + (−2.52 − 1.62i)3-s + (0.999 − 1.73i)4-s + (1.93 − 1.11i)5-s + (−4.23 − 0.198i)6-s + (3.85 − 5.84i)7-s − 2.82i·8-s + (3.75 + 8.18i)9-s + (1.58 − 2.73i)10-s + (−10.1 − 5.87i)11-s + (−5.33 + 2.75i)12-s + 0.292·13-s + (0.595 − 9.88i)14-s + (−6.70 − 0.314i)15-s + (−2.00 − 3.46i)16-s + (−25.0 − 14.4i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.841 − 0.540i)3-s + (0.249 − 0.433i)4-s + (0.387 − 0.223i)5-s + (−0.706 − 0.0331i)6-s + (0.551 − 0.834i)7-s − 0.353i·8-s + (0.416 + 0.909i)9-s + (0.158 − 0.273i)10-s + (−0.925 − 0.534i)11-s + (−0.444 + 0.229i)12-s + 0.0224·13-s + (0.0425 − 0.705i)14-s + (−0.446 − 0.0209i)15-s + (−0.125 − 0.216i)16-s + (−1.47 − 0.850i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 + 0.768i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.639 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.639 + 0.768i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.639 + 0.768i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.669756 - 1.42875i\)
\(L(\frac12)\) \(\approx\) \(0.669756 - 1.42875i\)
\(L(2)\) 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 \( 1 + (2.52 + 1.62i)T \)
5 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 + (-3.85 + 5.84i)T \)
good11 \( 1 + (10.1 + 5.87i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 0.292T + 169T^{2} \)
17 \( 1 + (25.0 + 14.4i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (3.00 + 5.20i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-27.2 + 15.7i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 27.0iT - 841T^{2} \)
31 \( 1 + (-1.74 + 3.02i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-27.5 - 47.6i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 23.5iT - 1.68e3T^{2} \)
43 \( 1 - 44.4T + 1.84e3T^{2} \)
47 \( 1 + (2.48 - 1.43i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-79.4 - 45.8i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-67.9 - 39.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (11.7 + 20.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-35.3 + 61.2i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 103. iT - 5.04e3T^{2} \)
73 \( 1 + (-33.8 + 58.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (65.2 + 113. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 50.5iT - 6.88e3T^{2} \)
89 \( 1 + (15.5 - 8.99i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 32.2T + 9.40e3T^{2} \)
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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

−11.71288828760992108182273743682, −10.93283837594316660794749787011, −10.39916690663737641654311442794, −8.803649401727849051798344721522, −7.41248145611043946508984798937, −6.52988525212841767306929257721, −5.24388354574412563512933858212, −4.53476904972338462130843649412, −2.49089426258228317926131544982, −0.813401854822189158559423182940, 2.34114327044439511218483233138, 4.18352552035262260520284981643, 5.23191601132047767937156540902, 5.98978586253445758955714478137, 7.12346705285566092214448818901, 8.508098065764041721795435767552, 9.652704607030146639156563277108, 10.84779019211353605780861032729, 11.42637765057263711763420442542, 12.61720855125443515956956121672

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