Properties

Label 2-354-59.58-c4-0-33
Degree $2$
Conductor $354$
Sign $-0.191 + 0.981i$
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 − 4.63·5-s + 14.6i·6-s − 33.3·7-s − 22.6i·8-s + 27·9-s − 13.1i·10-s − 4.16i·11-s − 41.5·12-s + 241. i·13-s − 94.2i·14-s − 24.0·15-s + 64.0·16-s + 291.·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577·3-s − 0.500·4-s − 0.185·5-s + 0.408i·6-s − 0.680·7-s − 0.353i·8-s + 0.333·9-s − 0.131i·10-s − 0.0344i·11-s − 0.288·12-s + 1.43i·13-s − 0.480i·14-s − 0.107·15-s + 0.250·16-s + 1.00·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.191 + 0.981i$
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: \((2,\ 354,\ (\ :2),\ -0.191 + 0.981i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.3628064280\)
\(L(\frac12)\) \(\approx\) \(0.3628064280\)
\(L(3)\) 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 \( 1 + (3.41e3 + 667. i)T \)
good5 \( 1 + 4.63T + 625T^{2} \)
7 \( 1 + 33.3T + 2.40e3T^{2} \)
11 \( 1 + 4.16iT - 1.46e4T^{2} \)
13 \( 1 - 241. iT - 2.85e4T^{2} \)
17 \( 1 - 291.T + 8.35e4T^{2} \)
19 \( 1 + 369.T + 1.30e5T^{2} \)
23 \( 1 + 496. iT - 2.79e5T^{2} \)
29 \( 1 + 329.T + 7.07e5T^{2} \)
31 \( 1 + 1.48e3iT - 9.23e5T^{2} \)
37 \( 1 + 727. iT - 1.87e6T^{2} \)
41 \( 1 + 1.74e3T + 2.82e6T^{2} \)
43 \( 1 + 1.68e3iT - 3.41e6T^{2} \)
47 \( 1 + 2.18e3iT - 4.87e6T^{2} \)
53 \( 1 + 2.22e3T + 7.89e6T^{2} \)
61 \( 1 - 4.20e3iT - 1.38e7T^{2} \)
67 \( 1 + 279. iT - 2.01e7T^{2} \)
71 \( 1 + 2.45e3T + 2.54e7T^{2} \)
73 \( 1 + 2.00e3iT - 2.83e7T^{2} \)
79 \( 1 - 3.62e3T + 3.89e7T^{2} \)
83 \( 1 - 2.78e3iT - 4.74e7T^{2} \)
89 \( 1 + 4.08e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.40e4iT - 8.85e7T^{2} \)
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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

−10.27106178561596689578695314793, −9.463735930741157225977616599895, −8.674996157098224389901107434750, −7.71958513547454524218685067878, −6.78104676572686352134196783489, −5.93370471565183078280192421793, −4.45085526902142947087988471112, −3.59170661644286481965459003545, −2.03368965589564736000226829042, −0.096099197597186156604388816813, 1.45222008787349342426416786434, 2.98765756408880730767966043889, 3.58563547573110218317587871935, 5.02090224798467708258759941723, 6.21210241153954546670666788605, 7.63382948592119311880911151228, 8.323128549643662665120491139310, 9.480434320018311506520357186167, 10.11424458512633432560036005156, 10.96446084593624909302707239441

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