Properties

Label 2-354-59.58-c4-0-6
Degree $2$
Conductor $354$
Sign $-0.953 - 0.301i$
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 + 6.35·5-s − 14.6i·6-s − 12.8·7-s − 22.6i·8-s + 27·9-s + 17.9i·10-s − 20.6i·11-s + 41.5·12-s − 24.6i·13-s − 36.4i·14-s − 33.0·15-s + 64.0·16-s + 132.·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577·3-s − 0.500·4-s + 0.254·5-s − 0.408i·6-s − 0.262·7-s − 0.353i·8-s + 0.333·9-s + 0.179i·10-s − 0.170i·11-s + 0.288·12-s − 0.145i·13-s − 0.185i·14-s − 0.146·15-s + 0.250·16-s + 0.458·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.301i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.953 - 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.953 - 0.301i$
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: \((2,\ 354,\ (\ :2),\ -0.953 - 0.301i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.8261946323\)
\(L(\frac12)\) \(\approx\) \(0.8261946323\)
\(L(3)\) 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 \( 1 + (1.05e3 - 3.31e3i)T \)
good5 \( 1 - 6.35T + 625T^{2} \)
7 \( 1 + 12.8T + 2.40e3T^{2} \)
11 \( 1 + 20.6iT - 1.46e4T^{2} \)
13 \( 1 + 24.6iT - 2.85e4T^{2} \)
17 \( 1 - 132.T + 8.35e4T^{2} \)
19 \( 1 - 240.T + 1.30e5T^{2} \)
23 \( 1 - 545. iT - 2.79e5T^{2} \)
29 \( 1 - 873.T + 7.07e5T^{2} \)
31 \( 1 + 984. iT - 9.23e5T^{2} \)
37 \( 1 - 2.55e3iT - 1.87e6T^{2} \)
41 \( 1 + 42.3T + 2.82e6T^{2} \)
43 \( 1 - 1.37e3iT - 3.41e6T^{2} \)
47 \( 1 + 691. iT - 4.87e6T^{2} \)
53 \( 1 + 3.28e3T + 7.89e6T^{2} \)
61 \( 1 - 1.15e3iT - 1.38e7T^{2} \)
67 \( 1 + 3.35e3iT - 2.01e7T^{2} \)
71 \( 1 + 7.96e3T + 2.54e7T^{2} \)
73 \( 1 - 7.26e3iT - 2.83e7T^{2} \)
79 \( 1 + 9.99e3T + 3.89e7T^{2} \)
83 \( 1 + 390. iT - 4.74e7T^{2} \)
89 \( 1 + 978. iT - 6.27e7T^{2} \)
97 \( 1 - 1.10e4iT - 8.85e7T^{2} \)
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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

−11.35908656186300151954527018966, −10.05060829589212578946143363887, −9.578591883527306447111712118554, −8.260017677295267605516203753735, −7.40464466982297894736664906576, −6.31133834853306821208453325111, −5.61960018204258332474048011179, −4.55598344311648645883352791600, −3.18491356549770966393592694786, −1.24109034150744936248754313013, 0.28682751491075065157201938502, 1.66220759751527897918797442839, 3.06026078454493572254023475156, 4.33900861986274097690523955665, 5.39432867605961952831962707295, 6.41608039830125113375160597179, 7.57783193509807251857342172592, 8.797455800784404651126334618568, 9.772005270383154479386354169753, 10.43500274305867337512406216052

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