Properties

Label 2-287-41.40-c1-0-13
Degree $2$
Conductor $287$
Sign $0.340 - 0.940i$
Analytic cond. $2.29170$
Root an. cond. $1.51383$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.08·2-s + 3.08i·3-s + 2.36·4-s + 1.51·5-s + 6.45i·6-s i·7-s + 0.758·8-s − 6.54·9-s + 3.15·10-s − 5.15i·11-s + 7.30i·12-s − 1.61i·13-s − 2.08i·14-s + 4.67i·15-s − 3.14·16-s − 1.19i·17-s + ⋯
L(s)  = 1  + 1.47·2-s + 1.78i·3-s + 1.18·4-s + 0.676·5-s + 2.63i·6-s − 0.377i·7-s + 0.268·8-s − 2.18·9-s + 0.998·10-s − 1.55i·11-s + 2.10i·12-s − 0.446i·13-s − 0.558i·14-s + 1.20i·15-s − 0.785·16-s − 0.288i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.340 - 0.940i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (204, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.340 - 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23202 + 1.56501i\)
\(L(\frac12)\) \(\approx\) \(2.23202 + 1.56501i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + iT \)
41 \( 1 + (6.01 + 2.18i)T \)
good2 \( 1 - 2.08T + 2T^{2} \)
3 \( 1 - 3.08iT - 3T^{2} \)
5 \( 1 - 1.51T + 5T^{2} \)
11 \( 1 + 5.15iT - 11T^{2} \)
13 \( 1 + 1.61iT - 13T^{2} \)
17 \( 1 + 1.19iT - 17T^{2} \)
19 \( 1 - 4.38iT - 19T^{2} \)
23 \( 1 - 7.15T + 23T^{2} \)
29 \( 1 - 7.89iT - 29T^{2} \)
31 \( 1 - 5.68T + 31T^{2} \)
37 \( 1 + 1.54T + 37T^{2} \)
43 \( 1 + 6.87T + 43T^{2} \)
47 \( 1 - 3.17iT - 47T^{2} \)
53 \( 1 + 4.55iT - 53T^{2} \)
59 \( 1 + 4.48T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 + 3.01iT - 67T^{2} \)
71 \( 1 + 14.5iT - 71T^{2} \)
73 \( 1 + 5.97T + 73T^{2} \)
79 \( 1 + 1.68iT - 79T^{2} \)
83 \( 1 - 5.09T + 83T^{2} \)
89 \( 1 - 11.4iT - 89T^{2} \)
97 \( 1 - 6.18iT - 97T^{2} \)
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

−11.93241191313974698761577976554, −11.01146843688265281218788104343, −10.40073434080553831713393655608, −9.346361809948376556300450366844, −8.431094892955786603415074910655, −6.44204476015642690219400343506, −5.47390811338822254679282068308, −4.94980169550660403089786744309, −3.62486663589357107820045973822, −3.10230173338348164352001177088, 1.89120383038228931579923198955, 2.73258548446856405763507099172, 4.63871828906052336581990974295, 5.68361006526356660360168992319, 6.65258267867314025363693529356, 7.13068398950223825871759313320, 8.538404502884837876169014039066, 9.738587835004306170086083534951, 11.47803622220731833922063336822, 11.96357844359355898928082722917

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