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Properties

Label 2-210-21.20-c3-0-12
Degree $2$
Conductor $210$
Sign $-0.731 - 0.681i$
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 + (3.71 + 3.63i)3-s − 4·4-s + 5·5-s + (−7.26 + 7.42i)6-s + (−0.448 − 18.5i)7-s − 8i·8-s + (0.598 + 26.9i)9-s + 10i·10-s + 37.8i·11-s + (−14.8 − 14.5i)12-s + 78.0i·13-s + (37.0 − 0.897i)14-s + (18.5 + 18.1i)15-s + 16·16-s − 28.9·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.714 + 0.699i)3-s − 0.5·4-s + 0.447·5-s + (−0.494 + 0.505i)6-s + (−0.0242 − 0.999i)7-s − 0.353i·8-s + (0.0221 + 0.999i)9-s + 0.316i·10-s + 1.03i·11-s + (−0.357 − 0.349i)12-s + 1.66i·13-s + (0.706 − 0.0171i)14-s + (0.319 + 0.312i)15-s + 0.250·16-s − 0.413·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.731 - 0.681i$
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: \((2,\ 210,\ (\ :3/2),\ -0.731 - 0.681i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.750516 + 1.90646i\)
\(L(\frac12)\) \(\approx\) \(0.750516 + 1.90646i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.71 - 3.63i)T \)
5 \( 1 - 5T \)
7 \( 1 + (0.448 + 18.5i)T \)
good11 \( 1 - 37.8iT - 1.33e3T^{2} \)
13 \( 1 - 78.0iT - 2.19e3T^{2} \)
17 \( 1 + 28.9T + 4.91e3T^{2} \)
19 \( 1 - 155. iT - 6.85e3T^{2} \)
23 \( 1 + 66.2iT - 1.21e4T^{2} \)
29 \( 1 + 166. iT - 2.43e4T^{2} \)
31 \( 1 + 62.4iT - 2.97e4T^{2} \)
37 \( 1 - 216.T + 5.06e4T^{2} \)
41 \( 1 - 247.T + 6.89e4T^{2} \)
43 \( 1 - 152.T + 7.95e4T^{2} \)
47 \( 1 + 133.T + 1.03e5T^{2} \)
53 \( 1 + 99.0iT - 1.48e5T^{2} \)
59 \( 1 + 460.T + 2.05e5T^{2} \)
61 \( 1 + 282. iT - 2.26e5T^{2} \)
67 \( 1 - 904.T + 3.00e5T^{2} \)
71 \( 1 + 611. iT - 3.57e5T^{2} \)
73 \( 1 - 1.02e3iT - 3.89e5T^{2} \)
79 \( 1 - 27.1T + 4.93e5T^{2} \)
83 \( 1 - 503.T + 5.71e5T^{2} \)
89 \( 1 - 129.T + 7.04e5T^{2} \)
97 \( 1 - 228. iT - 9.12e5T^{2} \)
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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

−12.54705249852787605865188963850, −11.06389723223508677116441456112, −9.872201746198665805972753372101, −9.536528213417863669489701548699, −8.228605179620306398128453904590, −7.31363919794917406590744955706, −6.20782007220254907856398001235, −4.56845401266748500909776153521, −3.99548500863215902026303887595, −1.96750999818397805170643882209, 0.822334679519620070179789740886, 2.50329808598597511878767610528, 3.20048570393306160640303388402, 5.21492895635418817117557711943, 6.27112950215324097633802314248, 7.76356368833596003976350869991, 8.798048869037071213204654019162, 9.310274633852470864728128308360, 10.70902031591403903496785301499, 11.59492434818563152854355353132

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