↑

Properties

Label 2-1305-145.144-c1-0-59
Degree $2$
Conductor $1305$
Sign $-0.734 + 0.678i$
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-s − 4-s + (0.280 − 2.21i)5-s + 3.46i·7-s − 3·8-s + (0.280 − 2.21i)10-s + 0.972i·11-s − 4.43i·13-s + 3.46i·14-s − 16-s − 2·17-s − 3.46i·19-s + (−0.280 + 2.21i)20-s + 0.972i·22-s − 3.46i·23-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.5·4-s + (0.125 − 0.992i)5-s + 1.30i·7-s − 1.06·8-s + (0.0887 − 0.701i)10-s + 0.293i·11-s − 1.23i·13-s + 0.925i·14-s − 0.250·16-s − 0.485·17-s − 0.794i·19-s + (−0.0627 + 0.496i)20-s + 0.207i·22-s − 0.722i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 + 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $-0.734 + 0.678i$
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: \((2,\ 1305,\ (\ :1/2),\ -0.734 + 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8887959658\)
\(L(\frac12)\) \(\approx\) \(0.8887959658\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.280 + 2.21i)T \)
29 \( 1 + (4.12 + 3.46i)T \)
good2 \( 1 - T + 2T^{2} \)
7 \( 1 - 3.46iT - 7T^{2} \)
11 \( 1 - 0.972iT - 11T^{2} \)
13 \( 1 + 4.43iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + 3.46iT - 23T^{2} \)
31 \( 1 + 7.90iT - 31T^{2} \)
37 \( 1 + 7.12T + 37T^{2} \)
41 \( 1 - 8.87iT - 41T^{2} \)
43 \( 1 + 5.43T + 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 + 4.43iT - 53T^{2} \)
59 \( 1 - 1.12T + 59T^{2} \)
61 \( 1 - 1.94iT - 61T^{2} \)
67 \( 1 + 12.3iT - 67T^{2} \)
71 \( 1 - 2.24T + 71T^{2} \)
73 \( 1 - 7.12T + 73T^{2} \)
79 \( 1 - 14.8iT - 79T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 - 8.87iT - 89T^{2} \)
97 \( 1 + 8.24T + 97T^{2} \)
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

−9.410359145687924732164836682948, −8.437882999601232196066081946763, −8.117682176864752141529975119799, −6.51569285147248066793496773061, −5.66939374135429044815198702286, −5.11962945969051842312858822497, −4.41564703159337459781153452433, −3.20763896773857598840161580957, −2.14819968035149791608343776060, −0.28452335447041207470936144583, 1.73127445152277369859802967010, 3.37889300479327093436981769400, 3.75756267574416350282101829013, 4.74999179007308745763572296139, 5.73915444991662464568455798372, 6.77204157401542369135781688130, 7.14901195118561412342305351381, 8.344330039227153852752416381318, 9.234134313774254958538745800171, 10.05304080368338921130622555501

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