↑

Properties

Label 2-1085-7.4-c1-0-46
Degree $2$
Conductor $1085$
Sign $0.654 + 0.755i$
Analytic cond. $8.66376$
Root an. cond. $2.94342$
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  + (−0.119 − 0.207i)2-s + (0.851 − 1.47i)3-s + (0.971 − 1.68i)4-s + (0.5 + 0.866i)5-s − 0.407·6-s + (0.663 + 2.56i)7-s − 0.944·8-s + (0.0500 + 0.0866i)9-s + (0.119 − 0.207i)10-s + (−2.59 + 4.49i)11-s + (−1.65 − 2.86i)12-s + 6.25·13-s + (0.451 − 0.444i)14-s + 1.70·15-s + (−1.82 − 3.16i)16-s + (2.78 − 4.82i)17-s + ⋯
L(s)  = 1  + (−0.0847 − 0.146i)2-s + (0.491 − 0.851i)3-s + (0.485 − 0.841i)4-s + (0.223 + 0.387i)5-s − 0.166·6-s + (0.250 + 0.968i)7-s − 0.333·8-s + (0.0166 + 0.0288i)9-s + (0.0378 − 0.0656i)10-s + (−0.783 + 1.35i)11-s + (−0.477 − 0.827i)12-s + 1.73·13-s + (0.120 − 0.118i)14-s + 0.439·15-s + (−0.457 − 0.792i)16-s + (0.675 − 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1085\)    =    \(5 \cdot 7 \cdot 31\)
Sign: $0.654 + 0.755i$
Analytic conductor: \(8.66376\)
Root analytic conductor: \(2.94342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1085} (466, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1085,\ (\ :1/2),\ 0.654 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.302066635\)
\(L(\frac12)\) \(\approx\) \(2.302066635\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.663 - 2.56i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.119 + 0.207i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.851 + 1.47i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (2.59 - 4.49i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.25T + 13T^{2} \)
17 \( 1 + (-2.78 + 4.82i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.55 + 6.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.80 - 3.12i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.27T + 29T^{2} \)
37 \( 1 + (1.69 + 2.93i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + (2.05 + 3.55i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.11 + 3.65i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.241 - 0.417i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.826 + 1.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.65 - 2.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.58T + 71T^{2} \)
73 \( 1 + (6.98 - 12.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.16 + 3.74i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + (-7.76 - 13.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.50T + 97T^{2} \)
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

−9.763383229471189555757457336015, −8.930844497661298796362213938235, −8.109068593154620855821627062594, −7.07601918621622116190097252075, −6.63152665175098512919097236458, −5.53042812730571100246846596035, −4.79178278591383429859522406228, −2.85277860786893876484872093914, −2.30037188897159991179473991265, −1.28876574392308449197352412211, 1.30013801002981638521463637695, 3.10534681251083147781926854416, 3.69678676994011386166136544826, 4.41263056398424120598145865103, 5.95487307193186248339640350755, 6.46013362950750547240694753443, 7.955105922087554544997236777546, 8.341152250061241025586137363147, 8.809264141427991738047625631965, 10.28362917733399237591155463593

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