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Properties

Label 2-1152-16.3-c2-0-18
Degree $2$
Conductor $1152$
Sign $0.829 - 0.558i$
Analytic cond. $31.3897$
Root an. cond. $5.60265$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.23 − 4.23i)5-s − 0.262·7-s + (8.60 + 8.60i)11-s + (15.9 + 15.9i)13-s − 3.51·17-s + (−10.7 + 10.7i)19-s − 16.4·23-s − 10.9i·25-s + (25.9 + 25.9i)29-s + 46.2i·31-s + (−1.11 + 1.11i)35-s + (2.99 − 2.99i)37-s − 21.9i·41-s + (−48.7 − 48.7i)43-s + 70.7i·47-s + ⋯
L(s)  = 1  + (0.847 − 0.847i)5-s − 0.0374·7-s + (0.782 + 0.782i)11-s + (1.23 + 1.23i)13-s − 0.206·17-s + (−0.566 + 0.566i)19-s − 0.717·23-s − 0.438i·25-s + (0.894 + 0.894i)29-s + 1.49i·31-s + (−0.0317 + 0.0317i)35-s + (0.0808 − 0.0808i)37-s − 0.534i·41-s + (−1.13 − 1.13i)43-s + 1.50i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.829 - 0.558i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ 0.829 - 0.558i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.382872015\)
\(L(\frac12)\) \(\approx\) \(2.382872015\)
\(L(2)\) 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 \( 1 + (-4.23 + 4.23i)T - 25iT^{2} \)
7 \( 1 + 0.262T + 49T^{2} \)
11 \( 1 + (-8.60 - 8.60i)T + 121iT^{2} \)
13 \( 1 + (-15.9 - 15.9i)T + 169iT^{2} \)
17 \( 1 + 3.51T + 289T^{2} \)
19 \( 1 + (10.7 - 10.7i)T - 361iT^{2} \)
23 \( 1 + 16.4T + 529T^{2} \)
29 \( 1 + (-25.9 - 25.9i)T + 841iT^{2} \)
31 \( 1 - 46.2iT - 961T^{2} \)
37 \( 1 + (-2.99 + 2.99i)T - 1.36e3iT^{2} \)
41 \( 1 + 21.9iT - 1.68e3T^{2} \)
43 \( 1 + (48.7 + 48.7i)T + 1.84e3iT^{2} \)
47 \( 1 - 70.7iT - 2.20e3T^{2} \)
53 \( 1 + (-52.8 + 52.8i)T - 2.80e3iT^{2} \)
59 \( 1 + (-61.7 - 61.7i)T + 3.48e3iT^{2} \)
61 \( 1 + (-22.9 - 22.9i)T + 3.72e3iT^{2} \)
67 \( 1 + (-54.9 + 54.9i)T - 4.48e3iT^{2} \)
71 \( 1 - 84.2T + 5.04e3T^{2} \)
73 \( 1 + 78.0iT - 5.32e3T^{2} \)
79 \( 1 + 59.2iT - 6.24e3T^{2} \)
83 \( 1 + (111. - 111. i)T - 6.88e3iT^{2} \)
89 \( 1 - 34.5iT - 7.92e3T^{2} \)
97 \( 1 + 66.0T + 9.40e3T^{2} \)
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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

−9.596349866403645443965989110373, −8.820489574897651274381269744652, −8.423312355722086829133342755884, −6.90671440186756316528823448150, −6.43373556682748301545726210618, −5.41540705374054390963704651773, −4.49521566075226514587901762240, −3.66059029058750503079145763193, −1.94584135549180347354621431387, −1.33778720214272434839939149529, 0.77026104392739033966977085232, 2.21744563911106737406980852430, 3.19454924245148315377360991925, 4.13318994647115046619593482150, 5.56636623221441462645762288341, 6.23896719097643193689081560721, 6.70452445501221981760998509765, 8.110539398348935173092478128540, 8.561165642203073693899056590753, 9.785482941076656968039723553664

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