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Properties

Label 2-1152-16.5-c1-0-7
Degree $2$
Conductor $1152$
Sign $0.923 + 0.382i$
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  + (−1 − i)5-s + 2i·7-s + (−1 − i)11-s + (1 − i)13-s + 2·17-s + (3 − 3i)19-s + 6i·23-s − 3i·25-s + (3 − 3i)29-s + 8·31-s + (2 − 2i)35-s + (−3 − 3i)37-s + (5 + 5i)43-s + 8·47-s + 3·49-s + ⋯
L(s)  = 1  + (−0.447 − 0.447i)5-s + 0.755i·7-s + (−0.301 − 0.301i)11-s + (0.277 − 0.277i)13-s + 0.485·17-s + (0.688 − 0.688i)19-s + 1.25i·23-s − 0.600i·25-s + (0.557 − 0.557i)29-s + 1.43·31-s + (0.338 − 0.338i)35-s + (−0.493 − 0.493i)37-s + (0.762 + 0.762i)43-s + 1.16·47-s + 0.428·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.923 + 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.495599919\)
\(L(\frac12)\) \(\approx\) \(1.495599919\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + i)T + 5iT^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + (1 + i)T + 11iT^{2} \)
13 \( 1 + (-1 + i)T - 13iT^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + (-3 + 3i)T - 19iT^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + (-3 + 3i)T - 29iT^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (3 + 3i)T + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-5 - 5i)T + 43iT^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + (5 + 5i)T + 53iT^{2} \)
59 \( 1 + (-3 - 3i)T + 59iT^{2} \)
61 \( 1 + (-9 + 9i)T - 61iT^{2} \)
67 \( 1 + (5 - 5i)T - 67iT^{2} \)
71 \( 1 + 10iT - 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-1 + i)T - 83iT^{2} \)
89 \( 1 + 4iT - 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−9.658496896538601961331976815719, −8.873291497111612933846721136141, −8.139892417713389916884358620360, −7.46307234656750136158883553530, −6.25636526933740798606536198205, −5.48036594909140966086025964907, −4.64393832626075969475875135623, −3.47647764324876054293851957952, −2.48332182477886635370329340956, −0.856868779798263474989688419630, 1.06727437927036061854231765705, 2.67381450202748859838066339126, 3.69832807006811576990550525925, 4.52825377038217125519558597020, 5.62414970792290860843230946545, 6.70570142232194579444992061677, 7.33673649050853434985665087803, 8.104087916944975258131967421443, 8.977261645849359773452334838514, 10.18467944199865474120508380945

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