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Properties

Label 2-354-177.176-c3-0-33
Degree $2$
Conductor $354$
Sign $0.998 + 0.0471i$
Analytic cond. $20.8866$
Root an. cond. $4.57019$
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  − 2·2-s + (5.02 + 1.30i)3-s + 4·4-s − 12.2i·5-s + (−10.0 − 2.60i)6-s + 29.7·7-s − 8·8-s + (23.5 + 13.1i)9-s + 24.5i·10-s + 28.5·11-s + (20.1 + 5.21i)12-s + 60.2i·13-s − 59.5·14-s + (16.0 − 61.7i)15-s + 16·16-s + 55.9i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.967 + 0.250i)3-s + 0.5·4-s − 1.09i·5-s + (−0.684 − 0.177i)6-s + 1.60·7-s − 0.353·8-s + (0.874 + 0.485i)9-s + 0.776i·10-s + 0.781·11-s + (0.483 + 0.125i)12-s + 1.28i·13-s − 1.13·14-s + (0.275 − 1.06i)15-s + 0.250·16-s + 0.798i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0471i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.998 + 0.0471i$
Analytic conductor: \(20.8866\)
Root analytic conductor: \(4.57019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :3/2),\ 0.998 + 0.0471i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.457441824\)
\(L(\frac12)\) \(\approx\) \(2.457441824\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2T \)
3 \( 1 + (-5.02 - 1.30i)T \)
59 \( 1 + (-443. + 92.8i)T \)
good5 \( 1 + 12.2iT - 125T^{2} \)
7 \( 1 - 29.7T + 343T^{2} \)
11 \( 1 - 28.5T + 1.33e3T^{2} \)
13 \( 1 - 60.2iT - 2.19e3T^{2} \)
17 \( 1 - 55.9iT - 4.91e3T^{2} \)
19 \( 1 + 75.7T + 6.85e3T^{2} \)
23 \( 1 - 54.0T + 1.21e4T^{2} \)
29 \( 1 + 189. iT - 2.43e4T^{2} \)
31 \( 1 + 279. iT - 2.97e4T^{2} \)
37 \( 1 - 82.3iT - 5.06e4T^{2} \)
41 \( 1 - 211. iT - 6.89e4T^{2} \)
43 \( 1 - 262. iT - 7.95e4T^{2} \)
47 \( 1 + 410.T + 1.03e5T^{2} \)
53 \( 1 - 436. iT - 1.48e5T^{2} \)
61 \( 1 + 745. iT - 2.26e5T^{2} \)
67 \( 1 - 317. iT - 3.00e5T^{2} \)
71 \( 1 + 61.7iT - 3.57e5T^{2} \)
73 \( 1 + 912. iT - 3.89e5T^{2} \)
79 \( 1 + 49.0T + 4.93e5T^{2} \)
83 \( 1 + 156.T + 5.71e5T^{2} \)
89 \( 1 - 901.T + 7.04e5T^{2} \)
97 \( 1 + 20.3iT - 9.12e5T^{2} \)
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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

−11.06519250778294128067104561988, −9.747500935952754104793962739153, −9.028645673194839389630392164001, −8.342805369811090711333120199528, −7.79907637467008770708292071875, −6.39781642759473806047653236713, −4.71534018516858188441833166820, −4.14009881989957535715036710669, −2.06420739295661275615666197696, −1.33149681711121113695789149064, 1.24668321270192302723050885995, 2.44191192497582521438538050849, 3.52727858734289413384497004479, 5.11796677005997301756217880507, 6.79210488545429050511126365406, 7.33735036315044663412688151172, 8.362183829064871487578597627222, 8.886962939860441311496097313509, 10.25887559589997052606451139068, 10.79825253237590593479235486712

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