↑

Properties

Label 2-354-177.176-c3-0-17
Degree $2$
Conductor $354$
Sign $0.964 - 0.265i$
Analytic cond. $20.8866$
Root an. cond. $4.57019$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + (−4.66 − 2.28i)3-s + 4·4-s − 8.40i·5-s + (9.33 + 4.56i)6-s + 31.0·7-s − 8·8-s + (16.5 + 21.3i)9-s + 16.8i·10-s + 8.36·11-s + (−18.6 − 9.13i)12-s + 33.8i·13-s − 62.0·14-s + (−19.1 + 39.2i)15-s + 16·16-s + 52.8i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.898 − 0.439i)3-s + 0.5·4-s − 0.751i·5-s + (0.635 + 0.310i)6-s + 1.67·7-s − 0.353·8-s + (0.614 + 0.789i)9-s + 0.531i·10-s + 0.229·11-s + (−0.449 − 0.219i)12-s + 0.722i·13-s − 1.18·14-s + (−0.330 + 0.674i)15-s + 0.250·16-s + 0.753i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(20.8866\)
Root analytic conductor: \(4.57019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :3/2),\ 0.964 - 0.265i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.114288135\)
\(L(\frac12)\) \(\approx\) \(1.114288135\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2T \)
3 \( 1 + (4.66 + 2.28i)T \)
59 \( 1 + (339. - 300. i)T \)
good5 \( 1 + 8.40iT - 125T^{2} \)
7 \( 1 - 31.0T + 343T^{2} \)
11 \( 1 - 8.36T + 1.33e3T^{2} \)
13 \( 1 - 33.8iT - 2.19e3T^{2} \)
17 \( 1 - 52.8iT - 4.91e3T^{2} \)
19 \( 1 + 13.4T + 6.85e3T^{2} \)
23 \( 1 + 124.T + 1.21e4T^{2} \)
29 \( 1 - 146. iT - 2.43e4T^{2} \)
31 \( 1 - 205. iT - 2.97e4T^{2} \)
37 \( 1 - 278. iT - 5.06e4T^{2} \)
41 \( 1 + 402. iT - 6.89e4T^{2} \)
43 \( 1 - 466. iT - 7.95e4T^{2} \)
47 \( 1 - 513.T + 1.03e5T^{2} \)
53 \( 1 - 100. iT - 1.48e5T^{2} \)
61 \( 1 - 413. iT - 2.26e5T^{2} \)
67 \( 1 + 771. iT - 3.00e5T^{2} \)
71 \( 1 - 262. iT - 3.57e5T^{2} \)
73 \( 1 + 355. iT - 3.89e5T^{2} \)
79 \( 1 - 942.T + 4.93e5T^{2} \)
83 \( 1 - 1.15e3T + 5.71e5T^{2} \)
89 \( 1 - 109.T + 7.04e5T^{2} \)
97 \( 1 + 1.05e3iT - 9.12e5T^{2} \)
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

−10.99292903601634731636074397364, −10.45790292442600752663063910326, −9.012599337932190930359985636603, −8.280812331071310862011386180216, −7.44507217317150696036907470539, −6.32862517562075853355116354532, −5.19376483619965628277384013307, −4.36493562265239586719400508648, −1.85158310321317023476768049045, −1.15247245179247763663723346562, 0.66707518401471919283818633756, 2.23022870521395341194359580364, 4.00493389266177729390263574805, 5.19836141481384038620691454098, 6.16510354950180817313487471015, 7.34461041771661784116891847639, 8.074790179896366440705940951035, 9.315510001046345234593348936997, 10.30644420864454463155329780716, 10.96127342016006580292176416359

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