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Properties

Label 2-1152-48.35-c3-0-41
Degree $2$
Conductor $1152$
Sign $-0.928 + 0.370i$
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  + (−3.43 − 3.43i)5-s + 8.14·7-s + (1.92 − 1.92i)11-s + (−9.16 − 9.16i)13-s + 28.6i·17-s + (−39.9 + 39.9i)19-s + 80.3i·23-s − 101. i·25-s + (113. − 113. i)29-s − 306. i·31-s + (−27.9 − 27.9i)35-s + (−47.8 + 47.8i)37-s − 349.·41-s + (164. + 164. i)43-s + 40.5·47-s + ⋯
L(s)  = 1  + (−0.307 − 0.307i)5-s + 0.439·7-s + (0.0526 − 0.0526i)11-s + (−0.195 − 0.195i)13-s + 0.409i·17-s + (−0.481 + 0.481i)19-s + 0.728i·23-s − 0.811i·25-s + (0.729 − 0.729i)29-s − 1.77i·31-s + (−0.135 − 0.135i)35-s + (−0.212 + 0.212i)37-s − 1.33·41-s + (0.583 + 0.583i)43-s + 0.125·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 + 0.370i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.928 + 0.370i$
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: \((2,\ 1152,\ (\ :3/2),\ -0.928 + 0.370i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5663692424\)
\(L(\frac12)\) \(\approx\) \(0.5663692424\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.43 + 3.43i)T + 125iT^{2} \)
7 \( 1 - 8.14T + 343T^{2} \)
11 \( 1 + (-1.92 + 1.92i)T - 1.33e3iT^{2} \)
13 \( 1 + (9.16 + 9.16i)T + 2.19e3iT^{2} \)
17 \( 1 - 28.6iT - 4.91e3T^{2} \)
19 \( 1 + (39.9 - 39.9i)T - 6.85e3iT^{2} \)
23 \( 1 - 80.3iT - 1.21e4T^{2} \)
29 \( 1 + (-113. + 113. i)T - 2.43e4iT^{2} \)
31 \( 1 + 306. iT - 2.97e4T^{2} \)
37 \( 1 + (47.8 - 47.8i)T - 5.06e4iT^{2} \)
41 \( 1 + 349.T + 6.89e4T^{2} \)
43 \( 1 + (-164. - 164. i)T + 7.95e4iT^{2} \)
47 \( 1 - 40.5T + 1.03e5T^{2} \)
53 \( 1 + (-454. - 454. i)T + 1.48e5iT^{2} \)
59 \( 1 + (245. - 245. i)T - 2.05e5iT^{2} \)
61 \( 1 + (186. + 186. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-118. + 118. i)T - 3.00e5iT^{2} \)
71 \( 1 + 414. iT - 3.57e5T^{2} \)
73 \( 1 - 431. iT - 3.89e5T^{2} \)
79 \( 1 + 1.04e3iT - 4.93e5T^{2} \)
83 \( 1 + (739. + 739. i)T + 5.71e5iT^{2} \)
89 \( 1 + 942.T + 7.04e5T^{2} \)
97 \( 1 + 983.T + 9.12e5T^{2} \)
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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

−8.988147957237844997443477900993, −8.119717415832380771193347774189, −7.67383196115424495361628234535, −6.45968941664708096714194933193, −5.69438791926234342982885223088, −4.61165779914972334830613648404, −3.92345196485016383285400145248, −2.63076969814912781677448313584, −1.45714066261081211564268344644, −0.13699667465809762045338147387, 1.34088271266699398002898067461, 2.58460807979046490371298775557, 3.59560779801853019072310251230, 4.69949955290619377798552099935, 5.37874120468069126393855720641, 6.77977354844649376992497246920, 7.06359831204100113170666684455, 8.295860326615919383637270697133, 8.792061132868426607738487131228, 9.832676304681570410017987510646

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