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Properties

Label 2-210-7.4-c3-0-5
Degree $2$
Conductor $210$
Sign $-0.773 - 0.633i$
Analytic cond. $12.3904$
Root an. cond. $3.52000$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (1.5 − 2.59i)3-s + (−1.99 + 3.46i)4-s + (2.5 + 4.33i)5-s + 6·6-s + (−10.8 + 15.0i)7-s − 7.99·8-s + (−4.5 − 7.79i)9-s + (−5 + 8.66i)10-s + (−12.9 + 22.3i)11-s + (6.00 + 10.3i)12-s − 2.85·13-s + (−36.8 − 3.68i)14-s + 15.0·15-s + (−8 − 13.8i)16-s + (−42.9 + 74.3i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s + 0.408·6-s + (−0.583 + 0.812i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 + 0.273i)10-s + (−0.354 + 0.613i)11-s + (0.144 + 0.249i)12-s − 0.0609·13-s + (−0.703 − 0.0703i)14-s + 0.258·15-s + (−0.125 − 0.216i)16-s + (−0.612 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 - 0.633i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.773 - 0.633i$
Analytic conductor: \(12.3904\)
Root analytic conductor: \(3.52000\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :3/2),\ -0.773 - 0.633i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.515798 + 1.44297i\)
\(L(\frac12)\) \(\approx\) \(0.515798 + 1.44297i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
3 \( 1 + (-1.5 + 2.59i)T \)
5 \( 1 + (-2.5 - 4.33i)T \)
7 \( 1 + (10.8 - 15.0i)T \)
good11 \( 1 + (12.9 - 22.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 2.85T + 2.19e3T^{2} \)
17 \( 1 + (42.9 - 74.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (7.73 + 13.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-30.1 - 52.2i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 29.6T + 2.43e4T^{2} \)
31 \( 1 + (126. - 219. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-102. - 177. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 61.8T + 6.89e4T^{2} \)
43 \( 1 + 65.9T + 7.95e4T^{2} \)
47 \( 1 + (87.7 + 152. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-200. + 347. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-196. + 341. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-30.4 - 52.7i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (56.9 - 98.6i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 483.T + 3.57e5T^{2} \)
73 \( 1 + (-28.8 + 49.9i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-509. - 882. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 870.T + 5.71e5T^{2} \)
89 \( 1 + (482. + 835. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 880.T + 9.12e5T^{2} \)
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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

−12.69439512972542953458155524762, −11.57171503842131859948877323906, −10.22432169855275153312234543701, −9.123526338865550290664533846602, −8.219180592439025218335279818125, −7.01373043977758972132349583124, −6.28004806575309218504272249814, −5.12142482764347127637446520601, −3.46470790548778947860713568262, −2.16266536538079544600407142675, 0.53583134612048889467506808526, 2.55112428346730330985558645896, 3.80606190393019456512081012823, 4.85091428285248487560898289389, 6.10823904864148618204778817458, 7.53196827517954980852188963306, 8.907293704340766111931583145709, 9.664556444589207943388576613474, 10.60984635646383452440890703695, 11.38107851734284162222799087369

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