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Properties

Label 2-1152-9.4-c1-0-46
Degree $2$
Conductor $1152$
Sign $-0.849 + 0.527i$
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  + (0.970 − 1.43i)3-s + (1.07 − 1.86i)5-s + (−0.153 − 0.265i)7-s + (−1.11 − 2.78i)9-s + (−2.50 − 4.34i)11-s + (0.470 − 0.815i)13-s + (−1.62 − 3.34i)15-s − 4.70·17-s − 1.61·19-s + (−0.529 − 0.0378i)21-s + (−4.08 + 7.06i)23-s + (0.191 + 0.330i)25-s + (−5.07 − 1.10i)27-s + (2.39 + 4.14i)29-s + (1.29 − 2.24i)31-s + ⋯
L(s)  = 1  + (0.560 − 0.828i)3-s + (0.480 − 0.832i)5-s + (−0.0578 − 0.100i)7-s + (−0.371 − 0.928i)9-s + (−0.755 − 1.30i)11-s + (0.130 − 0.226i)13-s + (−0.419 − 0.864i)15-s − 1.14·17-s − 0.371·19-s + (−0.115 − 0.00825i)21-s + (−0.851 + 1.47i)23-s + (0.0382 + 0.0661i)25-s + (−0.977 − 0.212i)27-s + (0.444 + 0.770i)29-s + (0.233 − 0.403i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.849 + 0.527i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.849 + 0.527i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.631784928\)
\(L(\frac12)\) \(\approx\) \(1.631784928\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.970 + 1.43i)T \)
good5 \( 1 + (-1.07 + 1.86i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.153 + 0.265i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.50 + 4.34i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.470 + 0.815i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.70T + 17T^{2} \)
19 \( 1 + 1.61T + 19T^{2} \)
23 \( 1 + (4.08 - 7.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.39 - 4.14i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.29 + 2.24i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 + (-3.86 + 6.69i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.138 + 0.239i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.92 + 3.32i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.23T + 53T^{2} \)
59 \( 1 + (-4.95 + 8.58i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.36 - 9.29i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.02 - 3.50i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.59T + 71T^{2} \)
73 \( 1 + 5.43T + 73T^{2} \)
79 \( 1 + (8.30 + 14.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.91 + 5.05i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.94T + 89T^{2} \)
97 \( 1 + (-7.07 - 12.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.170922542317322942659265212745, −8.624830735874806238813956715898, −7.981198352318704338444255568420, −7.05010673057282737131665295842, −5.98738579989377897107685560972, −5.45802224273477501291895967978, −4.08592207447968722461566402144, −2.97692927836730186330854199561, −1.90178865171048784554593043395, −0.62374736909922064500196638507, 2.37191840275917477427372019586, 2.58333344150051479664829976008, 4.24587774818090937746649881337, 4.64715418733043002182141483738, 6.01907562864365411541506101748, 6.74830410917953903057444752930, 7.80481995049696349726064768701, 8.538004582664432317082594056633, 9.542858887759356853203870194708, 10.08496525926414240874522278401

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