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Properties

Label 2-1152-48.35-c3-0-46
Degree $2$
Conductor $1152$
Sign $-0.992 + 0.125i$
Analytic cond. $67.9702$
Root an. cond. $8.24440$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.30 − 6.30i)5-s + 27.2·7-s + (4.03 − 4.03i)11-s + (−37.9 − 37.9i)13-s − 79.8i·17-s + (−75.2 + 75.2i)19-s − 25.2i·23-s − 45.3i·25-s + (107. − 107. i)29-s + 237. i·31-s + (−171. − 171. i)35-s + (−210. + 210. i)37-s − 378.·41-s + (−191. − 191. i)43-s + 417.·47-s + ⋯
L(s)  = 1  + (−0.564 − 0.564i)5-s + 1.46·7-s + (0.110 − 0.110i)11-s + (−0.809 − 0.809i)13-s − 1.13i·17-s + (−0.908 + 0.908i)19-s − 0.228i·23-s − 0.362i·25-s + (0.689 − 0.689i)29-s + 1.37i·31-s + (−0.828 − 0.828i)35-s + (−0.933 + 0.933i)37-s − 1.44·41-s + (−0.678 − 0.678i)43-s + 1.29·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.992 + 0.125i$
Analytic conductor: \(67.9702\)
Root analytic conductor: \(8.24440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ -0.992 + 0.125i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7376852567\)
\(L(\frac12)\) \(\approx\) \(0.7376852567\)
\(L(\frac{5}{2})\) 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 \( 1 + (6.30 + 6.30i)T + 125iT^{2} \)
7 \( 1 - 27.2T + 343T^{2} \)
11 \( 1 + (-4.03 + 4.03i)T - 1.33e3iT^{2} \)
13 \( 1 + (37.9 + 37.9i)T + 2.19e3iT^{2} \)
17 \( 1 + 79.8iT - 4.91e3T^{2} \)
19 \( 1 + (75.2 - 75.2i)T - 6.85e3iT^{2} \)
23 \( 1 + 25.2iT - 1.21e4T^{2} \)
29 \( 1 + (-107. + 107. i)T - 2.43e4iT^{2} \)
31 \( 1 - 237. iT - 2.97e4T^{2} \)
37 \( 1 + (210. - 210. i)T - 5.06e4iT^{2} \)
41 \( 1 + 378.T + 6.89e4T^{2} \)
43 \( 1 + (191. + 191. i)T + 7.95e4iT^{2} \)
47 \( 1 - 417.T + 1.03e5T^{2} \)
53 \( 1 + (139. + 139. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-282. + 282. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-255. - 255. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-348. + 348. i)T - 3.00e5iT^{2} \)
71 \( 1 - 321. iT - 3.57e5T^{2} \)
73 \( 1 - 135. iT - 3.89e5T^{2} \)
79 \( 1 + 522. iT - 4.93e5T^{2} \)
83 \( 1 + (444. + 444. i)T + 5.71e5iT^{2} \)
89 \( 1 + 1.10e3T + 7.04e5T^{2} \)
97 \( 1 + 1.06e3T + 9.12e5T^{2} \)
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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

−8.608063827179954746336096007424, −8.369451496274682261460286550895, −7.56403147205920663156375506938, −6.63179466540726720428056309605, −5.18778006675363491381990832980, −4.91857438363975060130388282597, −3.87119820701377642620225724980, −2.54727231515333734159699092039, −1.36276047255526904528913591628, −0.17419139062680795293445282084, 1.54056291206331570578669039789, 2.43837201964897566173531424976, 3.86279201547162161899111631475, 4.55441745227974078575455478630, 5.45687700559921177196537530157, 6.72639828557157694377300097327, 7.30093229869555615477894922187, 8.210217715435290998951179908940, 8.788150255795718982724829241897, 9.880014594296785564771432030568

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