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Properties

Label 2-1152-32.13-c1-0-10
Degree $2$
Conductor $1152$
Sign $0.980 + 0.195i$
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  + (3.12 − 1.29i)5-s + (−1 + i)7-s + (0.121 + 0.292i)11-s + (1.70 + 0.707i)13-s − 2.82i·17-s + (5.53 + 2.29i)19-s + (0.171 + 0.171i)23-s + (4.53 − 4.53i)25-s + (−1.12 + 2.70i)29-s + 4·31-s + (−1.82 + 4.41i)35-s + (1.70 − 0.707i)37-s + (5.82 + 5.82i)41-s + (−3.29 − 7.94i)43-s − 11.6i·47-s + ⋯
L(s)  = 1  + (1.39 − 0.578i)5-s + (−0.377 + 0.377i)7-s + (0.0365 + 0.0883i)11-s + (0.473 + 0.196i)13-s − 0.685i·17-s + (1.26 + 0.526i)19-s + (0.0357 + 0.0357i)23-s + (0.907 − 0.907i)25-s + (−0.208 + 0.502i)29-s + 0.718·31-s + (−0.309 + 0.746i)35-s + (0.280 − 0.116i)37-s + (0.910 + 0.910i)41-s + (−0.502 − 1.21i)43-s − 1.70i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.980 + 0.195i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.980 + 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.128146119\)
\(L(\frac12)\) \(\approx\) \(2.128146119\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.12 + 1.29i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (1 - i)T - 7iT^{2} \)
11 \( 1 + (-0.121 - 0.292i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-1.70 - 0.707i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 + (-5.53 - 2.29i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-0.171 - 0.171i)T + 23iT^{2} \)
29 \( 1 + (1.12 - 2.70i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-1.70 + 0.707i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-5.82 - 5.82i)T + 41iT^{2} \)
43 \( 1 + (3.29 + 7.94i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 11.6iT - 47T^{2} \)
53 \( 1 + (3.12 + 7.53i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (6.12 - 2.53i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-0.292 + 0.707i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (1.53 - 3.70i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (0.171 - 0.171i)T - 71iT^{2} \)
73 \( 1 + (-7 - 7i)T + 73iT^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 + (-6.12 - 2.53i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-2.65 + 2.65i)T - 89iT^{2} \)
97 \( 1 + 1.51T + 97T^{2} \)
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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

−9.639788301698135573987375475852, −9.178792297043674389820763534811, −8.320618619295921616758148719353, −7.18771852910160762704472174345, −6.24504914295008594772992172111, −5.56083427978972884646479591167, −4.85127007796923068943460270401, −3.44236516932480023170301866659, −2.31620501130018520575498621490, −1.18706530160206775151018304693, 1.24753920055481025693692017910, 2.54264168329440708203481312112, 3.43015518856683004835495000622, 4.72819062680994929194977626480, 5.88871003924765065431391411118, 6.26578050801755873565194577054, 7.23822424875164423983578894392, 8.156635777035908683012423118385, 9.395605222824713234504460074592, 9.659082425382646336893789053462

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