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Properties

Label 2-1152-32.21-c1-0-11
Degree $2$
Conductor $1152$
Sign $0.195 + 0.980i$
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  + (−1.12 + 2.70i)5-s + (−1 − i)7-s + (−4.12 − 1.70i)11-s + (0.292 + 0.707i)13-s − 2.82i·17-s + (−1.53 − 3.70i)19-s + (5.82 − 5.82i)23-s + (−2.53 − 2.53i)25-s + (3.12 − 1.29i)29-s + 4·31-s + (3.82 − 1.58i)35-s + (0.292 − 0.707i)37-s + (0.171 − 0.171i)41-s + (−4.70 − 1.94i)43-s + 0.343i·47-s + ⋯
L(s)  = 1  + (−0.501 + 1.21i)5-s + (−0.377 − 0.377i)7-s + (−1.24 − 0.514i)11-s + (0.0812 + 0.196i)13-s − 0.685i·17-s + (−0.352 − 0.850i)19-s + (1.21 − 1.21i)23-s + (−0.507 − 0.507i)25-s + (0.579 − 0.240i)29-s + 0.718·31-s + (0.647 − 0.268i)35-s + (0.0481 − 0.116i)37-s + (0.0267 − 0.0267i)41-s + (−0.717 − 0.297i)43-s + 0.0500i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.195 + 0.980i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.195 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8358977580\)
\(L(\frac12)\) \(\approx\) \(0.8358977580\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.12 - 2.70i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 + (4.12 + 1.70i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-0.292 - 0.707i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 + (1.53 + 3.70i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-5.82 + 5.82i)T - 23iT^{2} \)
29 \( 1 + (-3.12 + 1.29i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-0.292 + 0.707i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.171 + 0.171i)T - 41iT^{2} \)
43 \( 1 + (4.70 + 1.94i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 0.343iT - 47T^{2} \)
53 \( 1 + (-1.12 - 0.464i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (1.87 - 4.53i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-1.70 + 0.707i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-5.53 + 2.29i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (5.82 + 5.82i)T + 71iT^{2} \)
73 \( 1 + (-7 + 7i)T - 73iT^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 + (-1.87 - 4.53i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (8.65 + 8.65i)T + 89iT^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−9.792395708609121556448404339119, −8.695149685699372053200440930190, −7.915799875088937804896597275800, −6.92613936415673743160375247050, −6.64078033840874007841303405008, −5.31937647296716570949981876289, −4.34624790179702755383236326931, −3.07678376222119555791612015242, −2.64751007971841239694284996515, −0.38613089640302038840131141817, 1.29742766697671368267933058533, 2.74990769601051733156194435853, 3.91728883661682922292029970341, 4.94544858219513246933934420331, 5.50691490082748807937135468321, 6.63834548714558045257441208317, 7.80999536755305234848175534915, 8.249770192115618969620099884059, 9.092487522851617246251122871171, 9.921462199048382991038242366337

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