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Properties

Label 2-1152-72.59-c1-0-14
Degree $2$
Conductor $1152$
Sign $-0.299 - 0.954i$
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  + (0.765 + 1.55i)3-s + (0.781 − 1.35i)5-s + (−0.954 + 0.551i)7-s + (−1.82 + 2.37i)9-s + (−2.15 + 1.24i)11-s + (5.48 + 3.16i)13-s + (2.70 + 0.178i)15-s + 0.874i·17-s − 6.45·19-s + (−1.58 − 1.06i)21-s + (−1.52 + 2.64i)23-s + (1.27 + 2.21i)25-s + (−5.09 − 1.02i)27-s + (−0.767 − 1.32i)29-s + (6.29 + 3.63i)31-s + ⋯
L(s)  = 1  + (0.441 + 0.897i)3-s + (0.349 − 0.605i)5-s + (−0.360 + 0.208i)7-s + (−0.609 + 0.792i)9-s + (−0.649 + 0.374i)11-s + (1.52 + 0.879i)13-s + (0.697 + 0.0461i)15-s + 0.212i·17-s − 1.48·19-s + (−0.346 − 0.231i)21-s + (−0.318 + 0.552i)23-s + (0.255 + 0.442i)25-s + (−0.980 − 0.197i)27-s + (−0.142 − 0.246i)29-s + (1.13 + 0.652i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.299 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.299 - 0.954i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.299 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.641252556\)
\(L(\frac12)\) \(\approx\) \(1.641252556\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.765 - 1.55i)T \)
good5 \( 1 + (-0.781 + 1.35i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.954 - 0.551i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.15 - 1.24i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.48 - 3.16i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.874iT - 17T^{2} \)
19 \( 1 + 6.45T + 19T^{2} \)
23 \( 1 + (1.52 - 2.64i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.767 + 1.32i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.29 - 3.63i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.74iT - 37T^{2} \)
41 \( 1 + (-7.45 - 4.30i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.66 - 4.61i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.73 - 3.00i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 + (1.76 + 1.02i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.23 + 3.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.14 + 1.97i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.1T + 71T^{2} \)
73 \( 1 + 4.92T + 73T^{2} \)
79 \( 1 + (-3.31 + 1.91i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.2 - 6.47i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.3iT - 89T^{2} \)
97 \( 1 + (-4.65 - 8.05i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.827676878708437196832726789718, −9.271249470024941540487692185876, −8.552092420495056713979772918649, −7.927291581584769147071779040764, −6.48283002102165018990554103297, −5.79515462226630489480799829719, −4.69807163479309632953782453762, −4.05091806584765341326990178663, −2.90566515096509275854832218728, −1.69405130378860078423670043598, 0.66886138634671944956601798507, 2.24279308352788402821442307592, 3.04052508399667363719893213301, 4.05117401633276355401696912077, 5.67857939910646114832394666902, 6.31194039403166871492201915742, 6.91741941286760023664863186734, 8.205070606906437620516247477191, 8.331302637689283647701302301523, 9.498499699369418528521969533006

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