Properties

Label 2-1152-8.3-c4-0-5
Degree $2$
Conductor $1152$
Sign $-i$
Analytic cond. $119.082$
Root an. cond. $10.9124$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8i·5-s − 78.3i·7-s − 107.·11-s + 216i·13-s + 162·17-s − 440.·19-s − 705. i·23-s + 561·25-s − 1.30e3i·29-s + 627. i·31-s − 627.·35-s + 1.51e3i·37-s − 1.89e3·41-s − 2.90e3·43-s + 1.41e3i·47-s + ⋯
L(s)  = 1  − 0.320i·5-s − 1.59i·7-s − 0.890·11-s + 1.27i·13-s + 0.560·17-s − 1.22·19-s − 1.33i·23-s + 0.897·25-s − 1.55i·29-s + 0.652i·31-s − 0.511·35-s + 1.10i·37-s − 1.12·41-s − 1.57·43-s + 0.638i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-i$
Analytic conductor: \(119.082\)
Root analytic conductor: \(10.9124\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :2),\ -i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.5178802432\)
\(L(\frac12)\) \(\approx\) \(0.5178802432\)
\(L(3)\) 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 \( 1 + 8iT - 625T^{2} \)
7 \( 1 + 78.3iT - 2.40e3T^{2} \)
11 \( 1 + 107.T + 1.46e4T^{2} \)
13 \( 1 - 216iT - 2.85e4T^{2} \)
17 \( 1 - 162T + 8.35e4T^{2} \)
19 \( 1 + 440.T + 1.30e5T^{2} \)
23 \( 1 + 705. iT - 2.79e5T^{2} \)
29 \( 1 + 1.30e3iT - 7.07e5T^{2} \)
31 \( 1 - 627. iT - 9.23e5T^{2} \)
37 \( 1 - 1.51e3iT - 1.87e6T^{2} \)
41 \( 1 + 1.89e3T + 2.82e6T^{2} \)
43 \( 1 + 2.90e3T + 3.41e6T^{2} \)
47 \( 1 - 1.41e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.97e3iT - 7.89e6T^{2} \)
59 \( 1 + 2.26e3T + 1.21e7T^{2} \)
61 \( 1 + 2.37e3iT - 1.38e7T^{2} \)
67 \( 1 - 1.67e3T + 2.01e7T^{2} \)
71 \( 1 + 7.75e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.75e3T + 2.83e7T^{2} \)
79 \( 1 - 7.99e3iT - 3.89e7T^{2} \)
83 \( 1 - 9.33e3T + 4.74e7T^{2} \)
89 \( 1 + 2.43e3T + 6.27e7T^{2} \)
97 \( 1 - 7.45e3T + 8.85e7T^{2} \)
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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

−9.561343807405751036548565674886, −8.458883441224840682300163725368, −7.910852334402969508343094696686, −6.83435865464586927037342008683, −6.39674219921815140614559443301, −4.82098754982617094350144880767, −4.45249123042718133300085656995, −3.36485861475174179310827934315, −2.06104512877774804684490008831, −0.899375656574012555344506807654, 0.12113231724545790396343909530, 1.78213705990231868549985650750, 2.76590957787220486317759309251, 3.45717730873494792467102620800, 5.19224436626559963113623584177, 5.41024345491123009443467511594, 6.43014329700367344424045704164, 7.48576432565562526562510198249, 8.341459391786133969637377743354, 8.873352500699933731476441658564

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