Properties

Label 2-210-5.4-c3-0-12
Degree $2$
Conductor $210$
Sign $0.640 + 0.767i$
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 + (8.58 − 7.16i)5-s + 6·6-s + 7i·7-s + 8i·8-s − 9·9-s + (−14.3 − 17.1i)10-s + 30.6·11-s − 12i·12-s − 54.8i·13-s + 14·14-s + (21.4 + 25.7i)15-s + 16·16-s + 116. i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.767 − 0.640i)5-s + 0.408·6-s + 0.377i·7-s + 0.353i·8-s − 0.333·9-s + (−0.453 − 0.542i)10-s + 0.840·11-s − 0.288i·12-s − 1.16i·13-s + 0.267·14-s + (0.370 + 0.443i)15-s + 0.250·16-s + 1.66i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.640 + 0.767i$
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: \((2,\ 210,\ (\ :3/2),\ 0.640 + 0.767i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.79570 - 0.840085i\)
\(L(\frac12)\) \(\approx\) \(1.79570 - 0.840085i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-8.58 + 7.16i)T \)
7 \( 1 - 7iT \)
good11 \( 1 - 30.6T + 1.33e3T^{2} \)
13 \( 1 + 54.8iT - 2.19e3T^{2} \)
17 \( 1 - 116. iT - 4.91e3T^{2} \)
19 \( 1 - 105.T + 6.85e3T^{2} \)
23 \( 1 + 195. iT - 1.21e4T^{2} \)
29 \( 1 - 176.T + 2.43e4T^{2} \)
31 \( 1 - 159.T + 2.97e4T^{2} \)
37 \( 1 - 47.5iT - 5.06e4T^{2} \)
41 \( 1 - 83.8T + 6.89e4T^{2} \)
43 \( 1 + 280. iT - 7.95e4T^{2} \)
47 \( 1 + 538. iT - 1.03e5T^{2} \)
53 \( 1 - 555. iT - 1.48e5T^{2} \)
59 \( 1 + 494.T + 2.05e5T^{2} \)
61 \( 1 - 312.T + 2.26e5T^{2} \)
67 \( 1 - 1.06e3iT - 3.00e5T^{2} \)
71 \( 1 + 1.10e3T + 3.57e5T^{2} \)
73 \( 1 - 279. iT - 3.89e5T^{2} \)
79 \( 1 - 643.T + 4.93e5T^{2} \)
83 \( 1 - 578. iT - 5.71e5T^{2} \)
89 \( 1 - 117.T + 7.04e5T^{2} \)
97 \( 1 - 648. iT - 9.12e5T^{2} \)
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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

−11.94928036332232702494358319389, −10.48652370140049728255461176635, −10.08783734431758154857592263932, −8.897854734590767605525835531473, −8.310282702962232459473631959035, −6.26408209560378238230223438275, −5.29376823991481616178286800792, −4.13179153607396837746786202004, −2.69608329166519614258810054034, −1.06500562790745720517477535232, 1.32160384237932599394491304486, 3.12146957827705249853302875614, 4.82798584878994811954109720637, 6.12905566639893701644381612971, 6.93474756059210365853065720996, 7.64409001462231273293454499386, 9.300166182867803162648514154108, 9.651006477156212164706237461620, 11.28778346315728923826181285555, 11.96213460256418484216659271041

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