↑

Properties

Label 2-354-59.58-c4-0-23
Degree $2$
Conductor $354$
Sign $0.695 + 0.718i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·2-s − 5.19·3-s − 8.00·4-s + 21.2·5-s − 14.6i·6-s − 81.9·7-s − 22.6i·8-s + 27·9-s + 60.1i·10-s + 167. i·11-s + 41.5·12-s + 16.7i·13-s − 231. i·14-s − 110.·15-s + 64.0·16-s − 400.·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577·3-s − 0.500·4-s + 0.851·5-s − 0.408i·6-s − 1.67·7-s − 0.353i·8-s + 0.333·9-s + 0.601i·10-s + 1.38i·11-s + 0.288·12-s + 0.0991i·13-s − 1.18i·14-s − 0.491·15-s + 0.250·16-s − 1.38·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.695 + 0.718i$
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: \((2,\ 354,\ (\ :2),\ 0.695 + 0.718i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.5921648475\)
\(L(\frac12)\) \(\approx\) \(0.5921648475\)
\(L(3)\) 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 \( 1 + (-2.50e3 + 2.41e3i)T \)
good5 \( 1 - 21.2T + 625T^{2} \)
7 \( 1 + 81.9T + 2.40e3T^{2} \)
11 \( 1 - 167. iT - 1.46e4T^{2} \)
13 \( 1 - 16.7iT - 2.85e4T^{2} \)
17 \( 1 + 400.T + 8.35e4T^{2} \)
19 \( 1 - 475.T + 1.30e5T^{2} \)
23 \( 1 - 339. iT - 2.79e5T^{2} \)
29 \( 1 - 158.T + 7.07e5T^{2} \)
31 \( 1 + 1.04e3iT - 9.23e5T^{2} \)
37 \( 1 + 2.01e3iT - 1.87e6T^{2} \)
41 \( 1 + 1.22e3T + 2.82e6T^{2} \)
43 \( 1 + 1.63e3iT - 3.41e6T^{2} \)
47 \( 1 - 982. iT - 4.87e6T^{2} \)
53 \( 1 + 1.48e3T + 7.89e6T^{2} \)
61 \( 1 + 3.22e3iT - 1.38e7T^{2} \)
67 \( 1 - 2.60e3iT - 2.01e7T^{2} \)
71 \( 1 - 4.80e3T + 2.54e7T^{2} \)
73 \( 1 + 5.50e3iT - 2.83e7T^{2} \)
79 \( 1 - 1.22e4T + 3.89e7T^{2} \)
83 \( 1 + 4.32e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.50e4iT - 6.27e7T^{2} \)
97 \( 1 + 4.37e3iT - 8.85e7T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.40332537784864799556640074587, −9.506179989968655688733387050584, −9.348353870250315556919729659447, −7.49502041048661149701325160513, −6.73424513136567271198469930244, −6.03317983129680938946746354172, −5.05492035097208869814869241558, −3.76603583801644091031765155064, −2.12446962154942692265995231319, −0.22375904438177169475869022738, 0.973083119989565236878853102312, 2.66526632588327426789022070194, 3.57262026245758963230686289081, 5.12192285802516763175869795290, 6.13750595087297517507314599226, 6.73892361951145234338918908934, 8.498532366108596370376915130408, 9.433215123901679762037936649189, 10.05918405937990434659584762259, 10.89924862170593239688695430165

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