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Properties

Label 2-210-5.4-c3-0-1
Degree $2$
Conductor $210$
Sign $0.259 - 0.965i$
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 − 3i·3-s − 4·4-s + (−10.7 − 2.89i)5-s − 6·6-s − 7i·7-s + 8i·8-s − 9·9-s + (−5.79 + 21.5i)10-s − 13.5·11-s + 12i·12-s − 10.8i·13-s − 14·14-s + (−8.69 + 32.3i)15-s + 16·16-s + 98.7i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.965 − 0.259i)5-s − 0.408·6-s − 0.377i·7-s + 0.353i·8-s − 0.333·9-s + (−0.183 + 0.682i)10-s − 0.372·11-s + 0.288i·12-s − 0.230i·13-s − 0.267·14-s + (−0.149 + 0.557i)15-s + 0.250·16-s + 1.40i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.259 - 0.965i$
Analytic conductor: \(12.3904\)
Root analytic conductor: \(3.52000\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :3/2),\ 0.259 - 0.965i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.126344 + 0.0968980i\)
\(L(\frac12)\) \(\approx\) \(0.126344 + 0.0968980i\)
\(L(\frac{5}{2})\) 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 \( 1 + 3iT \)
5 \( 1 + (10.7 + 2.89i)T \)
7 \( 1 + 7iT \)
good11 \( 1 + 13.5T + 1.33e3T^{2} \)
13 \( 1 + 10.8iT - 2.19e3T^{2} \)
17 \( 1 - 98.7iT - 4.91e3T^{2} \)
19 \( 1 + 7.79T + 6.85e3T^{2} \)
23 \( 1 - 95.3iT - 1.21e4T^{2} \)
29 \( 1 + 0.202T + 2.43e4T^{2} \)
31 \( 1 + 165.T + 2.97e4T^{2} \)
37 \( 1 - 10.9iT - 5.06e4T^{2} \)
41 \( 1 - 12.3T + 6.89e4T^{2} \)
43 \( 1 + 358. iT - 7.95e4T^{2} \)
47 \( 1 - 450. iT - 1.03e5T^{2} \)
53 \( 1 - 286. iT - 1.48e5T^{2} \)
59 \( 1 + 739.T + 2.05e5T^{2} \)
61 \( 1 + 407.T + 2.26e5T^{2} \)
67 \( 1 - 77.8iT - 3.00e5T^{2} \)
71 \( 1 + 558.T + 3.57e5T^{2} \)
73 \( 1 + 28.8iT - 3.89e5T^{2} \)
79 \( 1 + 109.T + 4.93e5T^{2} \)
83 \( 1 + 1.26e3iT - 5.71e5T^{2} \)
89 \( 1 + 1.39e3T + 7.04e5T^{2} \)
97 \( 1 - 418. iT - 9.12e5T^{2} \)
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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

−12.23888278943848232028081976350, −11.16693954416084823653410521634, −10.50198184613798270623886284920, −9.095493781463820731722283256721, −8.096823550570734575065821451752, −7.33105876885253735161966102512, −5.78281987564497763226458096840, −4.34746701194927140644055397673, −3.24716889690416571304161461678, −1.49091904509365853135453919702, 0.06981912432493711037619914447, 2.95676985804368448748491716539, 4.30879966739248561189560399959, 5.27295449425948314989095103606, 6.66667252634248949789953071324, 7.66775032373958656470285886458, 8.640292286518087446431438172109, 9.571987058475427624377558485219, 10.76605587662133977452277012621, 11.66530183700855783739513311120

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