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Properties

Label 2-354-59.58-c6-0-56
Degree $2$
Conductor $354$
Sign $-0.103 + 0.994i$
Analytic cond. $81.4391$
Root an. cond. $9.02436$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.65i·2-s + 15.5·3-s − 32.0·4-s + 117.·5-s − 88.1i·6-s + 626.·7-s + 181. i·8-s + 243·9-s − 662. i·10-s − 2.44e3i·11-s − 498.·12-s + 657. i·13-s − 3.54e3i·14-s + 1.82e3·15-s + 1.02e3·16-s + 8.45e3·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577·3-s − 0.500·4-s + 0.936·5-s − 0.408i·6-s + 1.82·7-s + 0.353i·8-s + 0.333·9-s − 0.662i·10-s − 1.83i·11-s − 0.288·12-s + 0.299i·13-s − 1.29i·14-s + 0.540·15-s + 0.250·16-s + 1.72·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.103 + 0.994i$
Analytic conductor: \(81.4391\)
Root analytic conductor: \(9.02436\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :3),\ -0.103 + 0.994i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(4.101759862\)
\(L(\frac12)\) \(\approx\) \(4.101759862\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 5.65iT \)
3 \( 1 - 15.5T \)
59 \( 1 + (2.04e5 + 2.12e4i)T \)
good5 \( 1 - 117.T + 1.56e4T^{2} \)
7 \( 1 - 626.T + 1.17e5T^{2} \)
11 \( 1 + 2.44e3iT - 1.77e6T^{2} \)
13 \( 1 - 657. iT - 4.82e6T^{2} \)
17 \( 1 - 8.45e3T + 2.41e7T^{2} \)
19 \( 1 + 5.38e3T + 4.70e7T^{2} \)
23 \( 1 + 1.99e4iT - 1.48e8T^{2} \)
29 \( 1 + 2.34e4T + 5.94e8T^{2} \)
31 \( 1 - 1.13e3iT - 8.87e8T^{2} \)
37 \( 1 + 1.44e4iT - 2.56e9T^{2} \)
41 \( 1 + 1.41e4T + 4.75e9T^{2} \)
43 \( 1 - 1.15e5iT - 6.32e9T^{2} \)
47 \( 1 - 8.00e4iT - 1.07e10T^{2} \)
53 \( 1 - 1.98e5T + 2.21e10T^{2} \)
61 \( 1 + 1.10e5iT - 5.15e10T^{2} \)
67 \( 1 - 2.90e5iT - 9.04e10T^{2} \)
71 \( 1 - 2.43e5T + 1.28e11T^{2} \)
73 \( 1 + 4.44e5iT - 1.51e11T^{2} \)
79 \( 1 + 3.76e5T + 2.43e11T^{2} \)
83 \( 1 - 2.66e5iT - 3.26e11T^{2} \)
89 \( 1 - 1.17e5iT - 4.96e11T^{2} \)
97 \( 1 - 1.21e6iT - 8.32e11T^{2} \)
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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

−10.39981819111934570524889743019, −9.238397051126023798289744760734, −8.408744873438801903797465263119, −7.85726325040337574963174380462, −6.07754375896926469571999642007, −5.21791490215895128956615658857, −4.05219360455717698701589804783, −2.78754617750168539362505874157, −1.76500641133820174364546436680, −0.903689119124935976315429495891, 1.46579155681697390089936404754, 2.05342758311711086773538484922, 3.91263416476430365669481011620, 5.04774882573994434036627674327, 5.64146561851044293274375963029, 7.30840771321889029570287208346, 7.67396267290784625090178375732, 8.718758270490661065889454113726, 9.741417252538328556780324566648, 10.30528389442933632034660161467

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