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Properties

Label 2-1152-96.83-c1-0-12
Degree $2$
Conductor $1152$
Sign $-0.210 + 0.977i$
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  + (0.739 + 1.78i)5-s + (−0.385 − 0.385i)7-s + (−2.36 − 5.70i)11-s + (−2.30 − 0.956i)13-s − 5.05·17-s + (1.27 − 3.07i)19-s + (−2.28 − 2.28i)23-s + (0.892 − 0.892i)25-s + (−0.735 − 0.304i)29-s − 3.40i·31-s + (0.403 − 0.973i)35-s + (9.56 − 3.96i)37-s + (−5.27 + 5.27i)41-s + (−2.53 + 1.05i)43-s + 6.85i·47-s + ⋯
L(s)  = 1  + (0.330 + 0.798i)5-s + (−0.145 − 0.145i)7-s + (−0.712 − 1.72i)11-s + (−0.640 − 0.265i)13-s − 1.22·17-s + (0.292 − 0.705i)19-s + (−0.476 − 0.476i)23-s + (0.178 − 0.178i)25-s + (−0.136 − 0.0565i)29-s − 0.611i·31-s + (0.0681 − 0.164i)35-s + (1.57 − 0.651i)37-s + (−0.824 + 0.824i)41-s + (−0.386 + 0.160i)43-s + 1.00i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.210 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.210 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.210 + 0.977i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.210 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9583498571\)
\(L(\frac12)\) \(\approx\) \(0.9583498571\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.739 - 1.78i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.385 + 0.385i)T + 7iT^{2} \)
11 \( 1 + (2.36 + 5.70i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (2.30 + 0.956i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 5.05T + 17T^{2} \)
19 \( 1 + (-1.27 + 3.07i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (2.28 + 2.28i)T + 23iT^{2} \)
29 \( 1 + (0.735 + 0.304i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 3.40iT - 31T^{2} \)
37 \( 1 + (-9.56 + 3.96i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (5.27 - 5.27i)T - 41iT^{2} \)
43 \( 1 + (2.53 - 1.05i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 6.85iT - 47T^{2} \)
53 \( 1 + (-7.45 + 3.08i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-6.14 + 2.54i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-2.67 + 6.46i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (10.2 + 4.26i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (6.37 - 6.37i)T - 71iT^{2} \)
73 \( 1 + (-9.03 - 9.03i)T + 73iT^{2} \)
79 \( 1 + 1.22T + 79T^{2} \)
83 \( 1 + (14.8 + 6.14i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (4.97 + 4.97i)T + 89iT^{2} \)
97 \( 1 - 1.23T + 97T^{2} \)
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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

−9.687650695535099930813489179317, −8.676676851256270285986992006694, −7.983936096966825899226604276921, −6.95417119903729831328691865206, −6.25428146599606311324387202661, −5.43636782451697961233246905332, −4.30167108679816250974097928741, −3.03851051420665514429216065759, −2.43308530497786493682501975117, −0.39305672192717464097494683685, 1.64834761477451011530836636229, 2.56159310786532401422627820405, 4.15924884995910614299443118273, 4.87472449605212109521307451054, 5.62944011636172327035219551036, 6.84511361011379405671265351356, 7.48892277180501814413809869759, 8.483539827754859508577709314447, 9.319223794098781446440932264948, 9.901677017760639686234332504770

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