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Properties

Label 2-210-21.20-c3-0-1
Degree $2$
Conductor $210$
Sign $-0.897 + 0.441i$
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  + 2i·2-s + (−5.13 + 0.800i)3-s − 4·4-s + 5·5-s + (−1.60 − 10.2i)6-s + (5.52 + 17.6i)7-s − 8i·8-s + (25.7 − 8.21i)9-s + 10i·10-s + 11.6i·11-s + (20.5 − 3.20i)12-s + 81.5i·13-s + (−35.3 + 11.0i)14-s + (−25.6 + 4.00i)15-s + 16·16-s − 119.·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.988 + 0.154i)3-s − 0.5·4-s + 0.447·5-s + (−0.108 − 0.698i)6-s + (0.298 + 0.954i)7-s − 0.353i·8-s + (0.952 − 0.304i)9-s + 0.316i·10-s + 0.320i·11-s + (0.494 − 0.0770i)12-s + 1.73i·13-s + (−0.674 + 0.210i)14-s + (−0.441 + 0.0688i)15-s + 0.250·16-s − 1.71·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 + 0.441i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.897 + 0.441i$
Analytic conductor: \(12.3904\)
Root analytic conductor: \(3.52000\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :3/2),\ -0.897 + 0.441i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.135224 - 0.580745i\)
\(L(\frac12)\) \(\approx\) \(0.135224 - 0.580745i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
3 \( 1 + (5.13 - 0.800i)T \)
5 \( 1 - 5T \)
7 \( 1 + (-5.52 - 17.6i)T \)
good11 \( 1 - 11.6iT - 1.33e3T^{2} \)
13 \( 1 - 81.5iT - 2.19e3T^{2} \)
17 \( 1 + 119.T + 4.91e3T^{2} \)
19 \( 1 + 93.3iT - 6.85e3T^{2} \)
23 \( 1 + 126. iT - 1.21e4T^{2} \)
29 \( 1 - 46.8iT - 2.43e4T^{2} \)
31 \( 1 - 71.3iT - 2.97e4T^{2} \)
37 \( 1 + 392.T + 5.06e4T^{2} \)
41 \( 1 + 410.T + 6.89e4T^{2} \)
43 \( 1 - 167.T + 7.95e4T^{2} \)
47 \( 1 - 13.6T + 1.03e5T^{2} \)
53 \( 1 - 369. iT - 1.48e5T^{2} \)
59 \( 1 + 194.T + 2.05e5T^{2} \)
61 \( 1 + 490. iT - 2.26e5T^{2} \)
67 \( 1 - 389.T + 3.00e5T^{2} \)
71 \( 1 - 174. iT - 3.57e5T^{2} \)
73 \( 1 - 488. iT - 3.89e5T^{2} \)
79 \( 1 + 1.25e3T + 4.93e5T^{2} \)
83 \( 1 - 814.T + 5.71e5T^{2} \)
89 \( 1 - 1.30e3T + 7.04e5T^{2} \)
97 \( 1 - 1.12e3iT - 9.12e5T^{2} \)
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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

−12.39808337599587899129345085021, −11.54951647424427105590116775345, −10.60642096184145480194300054929, −9.247749842710836368070287888230, −8.772781232229358110828284360316, −6.84127940397328832741816430703, −6.52538515015433306964254210089, −5.12638958586181902918372165295, −4.45155965146495054346243052597, −1.99699555618688046857772165813, 0.28415294701461498369344529628, 1.68017825912046339493917676558, 3.62149004949785366940211487764, 4.94536523479389310590466191851, 5.92031644877787706067167448816, 7.22751346522496813921936748655, 8.364002051684554261204242007143, 9.908338149355885967907440616760, 10.53253156968508058737475036903, 11.20310690859944566539069954832

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