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Properties

Label 2-210-3.2-c2-0-0
Degree $2$
Conductor $210$
Sign $-0.503 + 0.863i$
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  + 1.41i·2-s + (−1.51 + 2.59i)3-s − 2.00·4-s + 2.23i·5-s + (−3.66 − 2.13i)6-s − 2.64·7-s − 2.82i·8-s + (−4.43 − 7.83i)9-s − 3.16·10-s + 14.6i·11-s + (3.02 − 5.18i)12-s − 19.1·13-s − 3.74i·14-s + (−5.79 − 3.37i)15-s + 4.00·16-s − 29.7i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.503 + 0.863i)3-s − 0.500·4-s + 0.447i·5-s + (−0.610 − 0.356i)6-s − 0.377·7-s − 0.353i·8-s + (−0.492 − 0.870i)9-s − 0.316·10-s + 1.33i·11-s + (0.251 − 0.431i)12-s − 1.47·13-s − 0.267i·14-s + (−0.386 − 0.225i)15-s + 0.250·16-s − 1.75i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.863i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.503 + 0.863i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.503 + 0.863i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.199632 - 0.347440i\)
\(L(\frac12)\) \(\approx\) \(0.199632 - 0.347440i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + (1.51 - 2.59i)T \)
5 \( 1 - 2.23iT \)
7 \( 1 + 2.64T \)
good11 \( 1 - 14.6iT - 121T^{2} \)
13 \( 1 + 19.1T + 169T^{2} \)
17 \( 1 + 29.7iT - 289T^{2} \)
19 \( 1 - 14.7T + 361T^{2} \)
23 \( 1 + 22.7iT - 529T^{2} \)
29 \( 1 - 51.5iT - 841T^{2} \)
31 \( 1 + 38.6T + 961T^{2} \)
37 \( 1 + 29.2T + 1.36e3T^{2} \)
41 \( 1 - 28.6iT - 1.68e3T^{2} \)
43 \( 1 + 40.4T + 1.84e3T^{2} \)
47 \( 1 - 10.5iT - 2.20e3T^{2} \)
53 \( 1 - 95.5iT - 2.80e3T^{2} \)
59 \( 1 + 11.1iT - 3.48e3T^{2} \)
61 \( 1 + 104.T + 3.72e3T^{2} \)
67 \( 1 - 45.2T + 4.48e3T^{2} \)
71 \( 1 - 93.7iT - 5.04e3T^{2} \)
73 \( 1 + 62.8T + 5.32e3T^{2} \)
79 \( 1 - 122.T + 6.24e3T^{2} \)
83 \( 1 - 68.2iT - 6.88e3T^{2} \)
89 \( 1 + 79.4iT - 7.92e3T^{2} \)
97 \( 1 + 28.2T + 9.40e3T^{2} \)
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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

−12.58327190873380057505765001566, −11.93256348365144305072089769946, −10.60579357475395763124275949025, −9.709909192138906673708141026151, −9.188231587565623064677891776085, −7.35508817901070300216969753463, −6.84331966552769950248307110323, −5.27873456142503152748852412271, −4.63594428411979415047730672844, −2.99394207117712309538951045503, 0.23025089808649407265884754722, 1.88534670401868889933467955498, 3.50994481399655168601698174009, 5.21032958406082309718859197495, 6.09202730224890948710932230543, 7.54683387487189015916436741914, 8.464696262663915033178496660077, 9.664877811366639683247378043918, 10.72119315929952898600439287053, 11.69543586132003481317407683901

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