↑

Properties

Label 2-1152-96.11-c1-0-6
Degree $2$
Conductor $1152$
Sign $0.920 - 0.391i$
Analytic cond. $9.19876$
Root an. cond. $3.03294$
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  + (2.97 + 1.23i)5-s + (−0.237 + 0.237i)7-s + (−2.12 − 0.879i)11-s + (0.0390 + 0.0943i)13-s + 4.16·17-s + (4.25 − 1.76i)19-s + (4.84 − 4.84i)23-s + (3.79 + 3.79i)25-s + (2.90 + 7.01i)29-s + 9.88i·31-s + (−0.999 + 0.414i)35-s + (0.175 − 0.424i)37-s + (−7.67 − 7.67i)41-s + (−2.99 + 7.23i)43-s − 6.10i·47-s + ⋯
L(s)  = 1  + (1.33 + 0.550i)5-s + (−0.0898 + 0.0898i)7-s + (−0.640 − 0.265i)11-s + (0.0108 + 0.0261i)13-s + 1.00·17-s + (0.975 − 0.404i)19-s + (1.01 − 1.01i)23-s + (0.758 + 0.758i)25-s + (0.539 + 1.30i)29-s + 1.77i·31-s + (−0.169 + 0.0700i)35-s + (0.0288 − 0.0697i)37-s + (−1.19 − 1.19i)41-s + (−0.456 + 1.10i)43-s − 0.891i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.920 - 0.391i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.920 - 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.104391261\)
\(L(\frac12)\) \(\approx\) \(2.104391261\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-2.97 - 1.23i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.237 - 0.237i)T - 7iT^{2} \)
11 \( 1 + (2.12 + 0.879i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-0.0390 - 0.0943i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 - 4.16T + 17T^{2} \)
19 \( 1 + (-4.25 + 1.76i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-4.84 + 4.84i)T - 23iT^{2} \)
29 \( 1 + (-2.90 - 7.01i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 9.88iT - 31T^{2} \)
37 \( 1 + (-0.175 + 0.424i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (7.67 + 7.67i)T + 41iT^{2} \)
43 \( 1 + (2.99 - 7.23i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 6.10iT - 47T^{2} \)
53 \( 1 + (-4.28 + 10.3i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (1.19 - 2.88i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-6.29 + 2.60i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-5.66 - 13.6i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (3.49 + 3.49i)T + 71iT^{2} \)
73 \( 1 + (1.42 - 1.42i)T - 73iT^{2} \)
79 \( 1 + 1.53T + 79T^{2} \)
83 \( 1 + (-2.95 - 7.14i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (1.92 - 1.92i)T - 89iT^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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

−10.10177417017136227055260760675, −9.065557153156979918089048817031, −8.373855829803814719795713622728, −7.09390441575220020673322336979, −6.62987759488819646406057598296, −5.41985742516281925731378632234, −5.11653286539103495782093496977, −3.30963421900022205804130582437, −2.65820005586373639003696100953, −1.30390676428896733556797129774, 1.11996899652323570547030512322, 2.27625955776930670661508517489, 3.42100785485080696928110433374, 4.81955399203720215372295114855, 5.52706828467956336572355851058, 6.14540414362035522021297406009, 7.36786149652057827504632376246, 8.063160471267673108637664253387, 9.148398830095802657068457250910, 9.881175323980672174293976572781

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