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Properties

Label 2-1152-9.7-c1-0-4
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: $\chi_{1152} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.849 - 0.527i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8868927205\)
\(L(\frac12)\) \(\approx\) \(0.8868927205\)
\(L(\frac{3}{2})\) 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

−10.09705038732219014226833395081, −9.140896378499461302085096751308, −8.748078010540045475966178983989, −7.70226661716153848612285893394, −7.16403059271504901433718392857, −5.56497353779480502627994580105, −4.75417335686177259959865615216, −4.25498017725036043342572613142, −3.04824967615828866066358919314, −1.84735309233810478524302517537, 0.33456715860255224070290922611, 2.19171085935502279658259870224, 2.98541320463179870794681140678, 3.89787421837436442224694855289, 5.29640010970421693213280224211, 6.44336627908160215902315666617, 6.91651871500394885817952928253, 7.83324714597523431968442701063, 8.591237740963540623371229050040, 9.111002467176229730509864489653

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