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Properties

Label 2-354-177.176-c3-0-25
Degree $2$
Conductor $354$
Sign $0.829 + 0.558i$
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 + (1.05 + 5.08i)3-s + 4·4-s − 6.49i·5-s + (−2.10 − 10.1i)6-s − 27.4·7-s − 8·8-s + (−24.7 + 10.6i)9-s + 12.9i·10-s + 15.6·11-s + (4.20 + 20.3i)12-s + 50.8i·13-s + 54.9·14-s + (33.0 − 6.82i)15-s + 16·16-s − 87.9i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.202 + 0.979i)3-s + 0.5·4-s − 0.580i·5-s + (−0.143 − 0.692i)6-s − 1.48·7-s − 0.353·8-s + (−0.918 + 0.396i)9-s + 0.410i·10-s + 0.430·11-s + (0.101 + 0.489i)12-s + 1.08i·13-s + 1.04·14-s + (0.568 − 0.117i)15-s + 0.250·16-s − 1.25i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.829 + 0.558i$
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: \((2,\ 354,\ (\ :3/2),\ 0.829 + 0.558i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8400985627\)
\(L(\frac12)\) \(\approx\) \(0.8400985627\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.05 - 5.08i)T \)
59 \( 1 + (-323. + 317. i)T \)
good5 \( 1 + 6.49iT - 125T^{2} \)
7 \( 1 + 27.4T + 343T^{2} \)
11 \( 1 - 15.6T + 1.33e3T^{2} \)
13 \( 1 - 50.8iT - 2.19e3T^{2} \)
17 \( 1 + 87.9iT - 4.91e3T^{2} \)
19 \( 1 + 54.3T + 6.85e3T^{2} \)
23 \( 1 - 138.T + 1.21e4T^{2} \)
29 \( 1 + 24.5iT - 2.43e4T^{2} \)
31 \( 1 + 176. iT - 2.97e4T^{2} \)
37 \( 1 + 128. iT - 5.06e4T^{2} \)
41 \( 1 + 27.5iT - 6.89e4T^{2} \)
43 \( 1 - 121. iT - 7.95e4T^{2} \)
47 \( 1 - 353.T + 1.03e5T^{2} \)
53 \( 1 + 249. iT - 1.48e5T^{2} \)
61 \( 1 + 441. iT - 2.26e5T^{2} \)
67 \( 1 + 148. iT - 3.00e5T^{2} \)
71 \( 1 + 259. iT - 3.57e5T^{2} \)
73 \( 1 - 229. iT - 3.89e5T^{2} \)
79 \( 1 + 4.13T + 4.93e5T^{2} \)
83 \( 1 - 1.17e3T + 5.71e5T^{2} \)
89 \( 1 + 305.T + 7.04e5T^{2} \)
97 \( 1 + 1.12e3iT - 9.12e5T^{2} \)
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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

−10.79086904314453759182838601621, −9.577781266413492268202370459083, −9.370495531006995679030213544045, −8.613785266532595048140620360655, −7.14137747660568766709261949208, −6.24531232799161759956796413781, −4.90638934382198703225343776361, −3.72576289940189200128596894180, −2.54209375684358257331615631785, −0.44250238301898684903760762830, 1.00641602752380751483206906805, 2.66084105236990467130170343674, 3.45060245700469171015094949695, 5.79981025718081462147558736769, 6.62147047980018430516566536229, 7.17955653708132845721735387211, 8.403489745047347920328571755645, 9.095571781991650171450641019923, 10.30298032744558716379764530245, 10.86043409186585577218949806692

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