↑

Properties

Label 2-1305-145.144-c1-0-26
Degree $2$
Conductor $1305$
Sign $1$
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  − 2·4-s − 2.23i·5-s + 5.38i·11-s + 4·16-s + 4.47i·20-s + 2.23i·23-s − 5.00·25-s − 5.38i·29-s + 12.0·37-s + 5.38i·41-s + 12.0·43-s − 10.7i·44-s + 7·49-s − 11.1i·53-s + 12.0·55-s + ⋯
L(s)  = 1  − 4-s − 0.999i·5-s + 1.62i·11-s + 16-s + 0.999i·20-s + 0.466i·23-s − 1.00·25-s − 0.999i·29-s + 1.97·37-s + 0.841i·41-s + 1.83·43-s − 1.62i·44-s + 49-s − 1.53i·53-s + 1.62·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $1$
Analytic conductor: \(10.4204\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1305,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.209199573\)
\(L(\frac12)\) \(\approx\) \(1.209199573\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23iT \)
29 \( 1 + 5.38iT \)
good2 \( 1 + 2T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 5.38iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 2.23iT - 23T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 12.0T + 37T^{2} \)
41 \( 1 - 5.38iT - 41T^{2} \)
43 \( 1 - 12.0T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 11.1iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 15.6iT - 83T^{2} \)
89 \( 1 + 10.7iT - 89T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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

−9.633707447013071776654572544945, −8.993300913265241440035247111510, −8.018932023320081428671109150551, −7.50530552887841372903272852926, −6.16531467087960261605465615692, −5.22943269081314089635970392005, −4.50528740654116124079001807500, −3.93652489556129409685887195005, −2.25347376882813811470605930560, −0.895306534938409028171532575126, 0.76028464924005956211439837042, 2.66394786707359359663027293418, 3.52605629396906719909437971818, 4.37871389256142542574020829343, 5.65496163549387476363382516234, 6.12100844234200891893254203097, 7.30089514700373161843594391373, 8.088307747936272021660608978913, 8.894354164122110518402517617050, 9.525580982329992271217552998696

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