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Properties

Label 2-354-59.58-c4-0-11
Degree $2$
Conductor $354$
Sign $0.0193 - 0.999i$
Analytic cond. $36.5929$
Root an. cond. $6.04921$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·2-s − 5.19·3-s − 8.00·4-s + 0.141·5-s − 14.6i·6-s − 89.1·7-s − 22.6i·8-s + 27·9-s + 0.400i·10-s − 153. i·11-s + 41.5·12-s − 17.2i·13-s − 252. i·14-s − 0.736·15-s + 64.0·16-s + 317.·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577·3-s − 0.500·4-s + 0.00566·5-s − 0.408i·6-s − 1.81·7-s − 0.353i·8-s + 0.333·9-s + 0.00400i·10-s − 1.26i·11-s + 0.288·12-s − 0.102i·13-s − 1.28i·14-s − 0.00327·15-s + 0.250·16-s + 1.10·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0193 - 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0193 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.0193 - 0.999i$
Analytic conductor: \(36.5929\)
Root analytic conductor: \(6.04921\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :2),\ 0.0193 - 0.999i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.6786882664\)
\(L(\frac12)\) \(\approx\) \(0.6786882664\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82iT \)
3 \( 1 + 5.19T \)
59 \( 1 + (3.48e3 + 67.4i)T \)
good5 \( 1 - 0.141T + 625T^{2} \)
7 \( 1 + 89.1T + 2.40e3T^{2} \)
11 \( 1 + 153. iT - 1.46e4T^{2} \)
13 \( 1 + 17.2iT - 2.85e4T^{2} \)
17 \( 1 - 317.T + 8.35e4T^{2} \)
19 \( 1 + 445.T + 1.30e5T^{2} \)
23 \( 1 + 279. iT - 2.79e5T^{2} \)
29 \( 1 - 307.T + 7.07e5T^{2} \)
31 \( 1 - 1.29e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.86e3iT - 1.87e6T^{2} \)
41 \( 1 - 718.T + 2.82e6T^{2} \)
43 \( 1 - 2.32e3iT - 3.41e6T^{2} \)
47 \( 1 - 662. iT - 4.87e6T^{2} \)
53 \( 1 - 1.41e3T + 7.89e6T^{2} \)
61 \( 1 - 1.56e3iT - 1.38e7T^{2} \)
67 \( 1 - 6.67e3iT - 2.01e7T^{2} \)
71 \( 1 - 2.26e3T + 2.54e7T^{2} \)
73 \( 1 + 407. iT - 2.83e7T^{2} \)
79 \( 1 + 1.33e3T + 3.89e7T^{2} \)
83 \( 1 - 1.00e4iT - 4.74e7T^{2} \)
89 \( 1 - 509. iT - 6.27e7T^{2} \)
97 \( 1 - 4.56e3iT - 8.85e7T^{2} \)
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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

−10.87583971678143046422243896292, −10.09180938142746287757839553487, −9.209627080696418516868272071858, −8.219833139649705087473319125391, −6.98714785695417476958140490855, −6.16107852521143401109314419494, −5.64966448820325933370185784464, −4.05085444908957309055561338904, −3.02452097949543301845844159116, −0.67399297073380613953468662403, 0.35561205342458310959478948809, 2.04235663227933878815311582538, 3.39456971115921298982842576163, 4.39125458085501548089300630912, 5.76661532285243112952977531385, 6.62006769182422144636840015955, 7.70252039460126831144949792778, 9.212735921588215463353089200592, 9.933202543710029671485390750784, 10.35433040238495736649539390942

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