↑

Properties

Label 2-1305-29.28-c1-0-28
Degree $2$
Conductor $1305$
Sign $0.964 + 0.262i$
Analytic cond. $10.4204$
Root an. cond. $3.22807$
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  − 0.517i·2-s + 1.73·4-s + 5-s + 0.732·7-s − 1.93i·8-s − 0.517i·10-s + 5.27i·11-s − 1.46·13-s − 0.378i·14-s + 2.46·16-s − 6.31i·17-s + 4.24i·19-s + 1.73·20-s + 2.73·22-s + 8.19·23-s + ⋯
L(s)  = 1  − 0.366i·2-s + 0.866·4-s + 0.447·5-s + 0.276·7-s − 0.683i·8-s − 0.163i·10-s + 1.59i·11-s − 0.406·13-s − 0.101i·14-s + 0.616·16-s − 1.53i·17-s + 0.973i·19-s + 0.387·20-s + 0.582·22-s + 1.70·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $0.964 + 0.262i$
Analytic conductor: \(10.4204\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1305,\ (\ :1/2),\ 0.964 + 0.262i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.395837520\)
\(L(\frac12)\) \(\approx\) \(2.395837520\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
29 \( 1 + (-5.19 - 1.41i)T \)
good2 \( 1 + 0.517iT - 2T^{2} \)
7 \( 1 - 0.732T + 7T^{2} \)
11 \( 1 - 5.27iT - 11T^{2} \)
13 \( 1 + 1.46T + 13T^{2} \)
17 \( 1 + 6.31iT - 17T^{2} \)
19 \( 1 - 4.24iT - 19T^{2} \)
23 \( 1 - 8.19T + 23T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 + 4.24iT - 37T^{2} \)
41 \( 1 + 8.76iT - 41T^{2} \)
43 \( 1 - 4.24iT - 43T^{2} \)
47 \( 1 - 8.38iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + 3.10iT - 61T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 1.13iT - 73T^{2} \)
79 \( 1 + 15.8iT - 79T^{2} \)
83 \( 1 + 2.19T + 83T^{2} \)
89 \( 1 + 2.07iT - 89T^{2} \)
97 \( 1 - 7.34iT - 97T^{2} \)
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

−9.707458107138268132647946684346, −9.090247201310059505531594213825, −7.72037653526070772012876409647, −7.15837899811132855352686026312, −6.51524986976371869489971150388, −5.27082211197452794400160972628, −4.60423143111177424650162524335, −3.16419386654685291055523922478, −2.32544833326335750309346553924, −1.32343538920174694371640696452, 1.19059846704778186101725998641, 2.51888163951511472130820667079, 3.34477138336643196267959678122, 4.80612519147083750021391927206, 5.69670812472436997251163372644, 6.38731141026002586145336221525, 7.05399038328580424735854791316, 8.243335259873889515227940987087, 8.530723771379059900775758159503, 9.696792571180074873440301512654

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