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Properties

Label 2-354-177.176-c3-0-0
Degree $2$
Conductor $354$
Sign $-0.0759 - 0.997i$
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 + (−5.11 − 0.917i)3-s + 4·4-s − 12.8i·5-s + (10.2 + 1.83i)6-s − 4.12·7-s − 8·8-s + (25.3 + 9.38i)9-s + 25.7i·10-s − 61.2·11-s + (−20.4 − 3.66i)12-s − 27.5i·13-s + 8.24·14-s + (−11.8 + 65.8i)15-s + 16·16-s − 98.6i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.984 − 0.176i)3-s + 0.5·4-s − 1.15i·5-s + (0.696 + 0.124i)6-s − 0.222·7-s − 0.353·8-s + (0.937 + 0.347i)9-s + 0.814i·10-s − 1.67·11-s + (−0.492 − 0.0882i)12-s − 0.587i·13-s + 0.157·14-s + (−0.203 + 1.13i)15-s + 0.250·16-s − 1.40i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0759 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0759 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.0759 - 0.997i$
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: \((2,\ 354,\ (\ :3/2),\ -0.0759 - 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.03183921098\)
\(L(\frac12)\) \(\approx\) \(0.03183921098\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.11 + 0.917i)T \)
59 \( 1 + (-113. - 438. i)T \)
good5 \( 1 + 12.8iT - 125T^{2} \)
7 \( 1 + 4.12T + 343T^{2} \)
11 \( 1 + 61.2T + 1.33e3T^{2} \)
13 \( 1 + 27.5iT - 2.19e3T^{2} \)
17 \( 1 + 98.6iT - 4.91e3T^{2} \)
19 \( 1 + 103.T + 6.85e3T^{2} \)
23 \( 1 - 124.T + 1.21e4T^{2} \)
29 \( 1 + 298. iT - 2.43e4T^{2} \)
31 \( 1 - 143. iT - 2.97e4T^{2} \)
37 \( 1 - 183. iT - 5.06e4T^{2} \)
41 \( 1 - 203. iT - 6.89e4T^{2} \)
43 \( 1 + 20.4iT - 7.95e4T^{2} \)
47 \( 1 + 219.T + 1.03e5T^{2} \)
53 \( 1 - 212. iT - 1.48e5T^{2} \)
61 \( 1 + 314. iT - 2.26e5T^{2} \)
67 \( 1 - 641. iT - 3.00e5T^{2} \)
71 \( 1 + 323. iT - 3.57e5T^{2} \)
73 \( 1 - 1.24e3iT - 3.89e5T^{2} \)
79 \( 1 - 735.T + 4.93e5T^{2} \)
83 \( 1 - 424.T + 5.71e5T^{2} \)
89 \( 1 - 1.01e3T + 7.04e5T^{2} \)
97 \( 1 - 1.76e3iT - 9.12e5T^{2} \)
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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

−11.18414201003628641649680282122, −10.34809774384095930839557159849, −9.570703681313020775229832314146, −8.398224224553378585193987582316, −7.65500142707445343925768654892, −6.51923470104350534917289492729, −5.33301927025013566543489331326, −4.72569765054422583176597877825, −2.60255122550836677369890642938, −0.909290851057909871494734592586, 0.02046430759356534128568305601, 2.02796880400388147511749094317, 3.45109938824264120493699518759, 5.02147263521600963133093402765, 6.22179926689198503180736757432, 6.87135300341527759227572112148, 7.83800042399547453705686136352, 9.070811255025261017474002847720, 10.35646732234777310246801219251, 10.65329600629453274259440514619

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