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Properties

Label 2-1152-96.83-c1-0-13
Degree $2$
Conductor $1152$
Sign $-0.943 + 0.330i$
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  + (1.11 + 2.70i)5-s + (−1.57 − 1.57i)7-s + (−1.00 − 2.41i)11-s + (−6.48 − 2.68i)13-s − 0.520·17-s + (−2.95 + 7.14i)19-s + (−1.35 − 1.35i)23-s + (−2.52 + 2.52i)25-s + (−5.31 − 2.20i)29-s + 1.54i·31-s + (2.48 − 6.00i)35-s + (−3.79 + 1.57i)37-s + (−1.08 + 1.08i)41-s + (2.71 − 1.12i)43-s − 11.1i·47-s + ⋯
L(s)  = 1  + (0.500 + 1.20i)5-s + (−0.593 − 0.593i)7-s + (−0.301 − 0.729i)11-s + (−1.79 − 0.745i)13-s − 0.126·17-s + (−0.678 + 1.63i)19-s + (−0.282 − 0.282i)23-s + (−0.504 + 0.504i)25-s + (−0.986 − 0.408i)29-s + 0.277i·31-s + (0.420 − 1.01i)35-s + (−0.624 + 0.258i)37-s + (−0.170 + 0.170i)41-s + (0.413 − 0.171i)43-s − 1.63i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.943 + 0.330i$
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: \((2,\ 1152,\ (\ :1/2),\ -0.943 + 0.330i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.03618488497\)
\(L(\frac12)\) \(\approx\) \(0.03618488497\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 - 2.70i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (1.57 + 1.57i)T + 7iT^{2} \)
11 \( 1 + (1.00 + 2.41i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (6.48 + 2.68i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 0.520T + 17T^{2} \)
19 \( 1 + (2.95 - 7.14i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (1.35 + 1.35i)T + 23iT^{2} \)
29 \( 1 + (5.31 + 2.20i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 1.54iT - 31T^{2} \)
37 \( 1 + (3.79 - 1.57i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.08 - 1.08i)T - 41iT^{2} \)
43 \( 1 + (-2.71 + 1.12i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 11.1iT - 47T^{2} \)
53 \( 1 + (-3.70 + 1.53i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-3.19 + 1.32i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (3.78 - 9.12i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (10.9 + 4.53i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (6.83 - 6.83i)T - 71iT^{2} \)
73 \( 1 + (-2.94 - 2.94i)T + 73iT^{2} \)
79 \( 1 + 8.79T + 79T^{2} \)
83 \( 1 + (-13.9 - 5.79i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (7.09 + 7.09i)T + 89iT^{2} \)
97 \( 1 + 5.91T + 97T^{2} \)
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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

−9.858347026496154473630957446558, −8.554374893411327156820446742324, −7.58841819567347169174538541704, −6.98474714787726397817860257411, −6.09978923307513423160396818746, −5.35713136694883384970025117065, −3.94304974656093544341546935370, −3.05490545034263823192016786871, −2.15252969936748118070637653414, −0.01402019850861008387141891062, 1.87795273758327847217842819486, 2.71289401234853364968739457350, 4.44545934768530798872206502202, 4.91809485048425244955586132518, 5.82313212416009449084922462439, 6.90801134651094516504108457071, 7.61468674075834815774282955959, 8.921453785114273627611934347004, 9.269760336526124937415389330968, 9.812251873550941025440569485259

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