Properties

Label 2-1152-96.83-c1-0-9
Degree $2$
Conductor $1152$
Sign $0.739 + 0.673i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0963 − 0.232i)5-s + (−0.617 − 0.617i)7-s + (−0.505 − 1.21i)11-s + (3.41 + 1.41i)13-s + 2.76·17-s + (−0.189 + 0.458i)19-s + (−4.46 − 4.46i)23-s + (3.49 − 3.49i)25-s + (−0.0101 − 0.00418i)29-s + 4.03i·31-s + (−0.0841 + 0.203i)35-s + (6.30 − 2.61i)37-s + (5.34 − 5.34i)41-s + (10.1 − 4.21i)43-s − 11.5i·47-s + ⋯
L(s)  = 1  + (−0.0430 − 0.104i)5-s + (−0.233 − 0.233i)7-s + (−0.152 − 0.367i)11-s + (0.947 + 0.392i)13-s + 0.669·17-s + (−0.0435 + 0.105i)19-s + (−0.931 − 0.931i)23-s + (0.698 − 0.698i)25-s + (−0.00187 − 0.000777i)29-s + 0.724i·31-s + (−0.0142 + 0.0343i)35-s + (1.03 − 0.429i)37-s + (0.834 − 0.834i)41-s + (1.55 − 0.642i)43-s − 1.69i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.739 + 0.673i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.739 + 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.570423736\)
\(L(\frac12)\) \(\approx\) \(1.570423736\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0963 + 0.232i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.617 + 0.617i)T + 7iT^{2} \)
11 \( 1 + (0.505 + 1.21i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-3.41 - 1.41i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 2.76T + 17T^{2} \)
19 \( 1 + (0.189 - 0.458i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (4.46 + 4.46i)T + 23iT^{2} \)
29 \( 1 + (0.0101 + 0.00418i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 4.03iT - 31T^{2} \)
37 \( 1 + (-6.30 + 2.61i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-5.34 + 5.34i)T - 41iT^{2} \)
43 \( 1 + (-10.1 + 4.21i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 11.5iT - 47T^{2} \)
53 \( 1 + (9.04 - 3.74i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (0.939 - 0.389i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (2.97 - 7.17i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-7.40 - 3.06i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-1.20 + 1.20i)T - 71iT^{2} \)
73 \( 1 + (-3.73 - 3.73i)T + 73iT^{2} \)
79 \( 1 - 7.22T + 79T^{2} \)
83 \( 1 + (-11.2 - 4.67i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (3.70 + 3.70i)T + 89iT^{2} \)
97 \( 1 + 14.0T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.732694984747034288397948853958, −8.768032533898371957027514861644, −8.207260238497759478713966754006, −7.19111387288078481763649540551, −6.30384437152980704271674842832, −5.57513452573663909203994101687, −4.35737146944742596065616105100, −3.57592796596523789513240047224, −2.33691200221825026405932244871, −0.807937165998463373448341439737, 1.23788732589081133982322691218, 2.70181443723446105534857556939, 3.65095664925916677934259906312, 4.71376972075880841330904819509, 5.83412416552815764474199466191, 6.35723897725603629118028355059, 7.70646991426291974748064848090, 7.987108201376198381715550511409, 9.385859662211237077044158799982, 9.605457696443513037905592412136

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