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Properties

Label 2-354-177.176-c5-0-29
Degree $2$
Conductor $354$
Sign $0.845 + 0.534i$
Analytic cond. $56.7758$
Root an. cond. $7.53497$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + (−0.499 − 15.5i)3-s + 16·4-s − 12.6i·5-s + (1.99 + 62.3i)6-s − 26.2·7-s − 64·8-s + (−242. + 15.5i)9-s + 50.7i·10-s − 726.·11-s + (−7.99 − 249. i)12-s + 833. i·13-s + 105.·14-s + (−197. + 6.33i)15-s + 256·16-s − 152. i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.0320 − 0.999i)3-s + 0.5·4-s − 0.226i·5-s + (0.0226 + 0.706i)6-s − 0.202·7-s − 0.353·8-s + (−0.997 + 0.0640i)9-s + 0.160i·10-s − 1.81·11-s + (−0.0160 − 0.499i)12-s + 1.36i·13-s + 0.143·14-s + (−0.226 + 0.00726i)15-s + 0.250·16-s − 0.128i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.534i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.845 + 0.534i$
Analytic conductor: \(56.7758\)
Root analytic conductor: \(7.53497\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :5/2),\ 0.845 + 0.534i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8137181915\)
\(L(\frac12)\) \(\approx\) \(0.8137181915\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4T \)
3 \( 1 + (0.499 + 15.5i)T \)
59 \( 1 + (-1.50e4 + 2.21e4i)T \)
good5 \( 1 + 12.6iT - 3.12e3T^{2} \)
7 \( 1 + 26.2T + 1.68e4T^{2} \)
11 \( 1 + 726.T + 1.61e5T^{2} \)
13 \( 1 - 833. iT - 3.71e5T^{2} \)
17 \( 1 + 152. iT - 1.41e6T^{2} \)
19 \( 1 + 1.12e3T + 2.47e6T^{2} \)
23 \( 1 - 1.57e3T + 6.43e6T^{2} \)
29 \( 1 + 5.29e3iT - 2.05e7T^{2} \)
31 \( 1 + 3.98e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.28e4iT - 6.93e7T^{2} \)
41 \( 1 + 9.85e3iT - 1.15e8T^{2} \)
43 \( 1 - 7.64e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.30e4T + 2.29e8T^{2} \)
53 \( 1 - 3.56e4iT - 4.18e8T^{2} \)
61 \( 1 + 2.27e4iT - 8.44e8T^{2} \)
67 \( 1 + 8.75e3iT - 1.35e9T^{2} \)
71 \( 1 + 1.84e4iT - 1.80e9T^{2} \)
73 \( 1 + 504. iT - 2.07e9T^{2} \)
79 \( 1 + 3.86e4T + 3.07e9T^{2} \)
83 \( 1 - 2.23e4T + 3.93e9T^{2} \)
89 \( 1 - 1.01e5T + 5.58e9T^{2} \)
97 \( 1 + 2.02e4iT - 8.58e9T^{2} \)
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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

−10.63616691179134810258830554803, −9.496211602963350840426999309717, −8.543365369795368510368301698125, −7.80355380563521618174907843661, −6.91014867011290078802851464537, −6.01616033526140351727633623752, −4.75901082732936237751963793737, −2.84970492802174851113728428374, −1.96390339570154816730299035861, −0.57560863645253091184166505050, 0.47099092157024284006271894353, 2.56119465287011288908991216606, 3.32416969108754293593218103897, 4.98241694682108617385228295702, 5.69925771865538403129589498824, 7.10628843781303792226783531431, 8.177882225690040171045957417545, 8.843171337002878290251747513137, 10.06105310331899801896713926250, 10.59062078685997837228604617901

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