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Properties

Label 2-1152-16.11-c2-0-3
Degree $2$
Conductor $1152$
Sign $-0.835 - 0.548i$
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  + (−0.0586 − 0.0586i)5-s + 4.61·7-s + (−5.36 + 5.36i)11-s + (−11.0 + 11.0i)13-s + 12.8·17-s + (−2.63 − 2.63i)19-s − 16.3·23-s − 24.9i·25-s + (−26.0 + 26.0i)29-s + 20.2i·31-s + (−0.270 − 0.270i)35-s + (−41.2 − 41.2i)37-s − 3.29i·41-s + (0.786 − 0.786i)43-s + 79.7i·47-s + ⋯
L(s)  = 1  + (−0.0117 − 0.0117i)5-s + 0.659·7-s + (−0.487 + 0.487i)11-s + (−0.850 + 0.850i)13-s + 0.757·17-s + (−0.138 − 0.138i)19-s − 0.712·23-s − 0.999i·25-s + (−0.898 + 0.898i)29-s + 0.652i·31-s + (−0.00773 − 0.00773i)35-s + (−1.11 − 1.11i)37-s − 0.0804i·41-s + (0.0183 − 0.0183i)43-s + 1.69i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.548i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.835 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.835 - 0.548i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ -0.835 - 0.548i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7041596722\)
\(L(\frac12)\) \(\approx\) \(0.7041596722\)
\(L(2)\) 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 \( 1 + (0.0586 + 0.0586i)T + 25iT^{2} \)
7 \( 1 - 4.61T + 49T^{2} \)
11 \( 1 + (5.36 - 5.36i)T - 121iT^{2} \)
13 \( 1 + (11.0 - 11.0i)T - 169iT^{2} \)
17 \( 1 - 12.8T + 289T^{2} \)
19 \( 1 + (2.63 + 2.63i)T + 361iT^{2} \)
23 \( 1 + 16.3T + 529T^{2} \)
29 \( 1 + (26.0 - 26.0i)T - 841iT^{2} \)
31 \( 1 - 20.2iT - 961T^{2} \)
37 \( 1 + (41.2 + 41.2i)T + 1.36e3iT^{2} \)
41 \( 1 + 3.29iT - 1.68e3T^{2} \)
43 \( 1 + (-0.786 + 0.786i)T - 1.84e3iT^{2} \)
47 \( 1 - 79.7iT - 2.20e3T^{2} \)
53 \( 1 + (-1.06 - 1.06i)T + 2.80e3iT^{2} \)
59 \( 1 + (-32.5 + 32.5i)T - 3.48e3iT^{2} \)
61 \( 1 + (15.2 - 15.2i)T - 3.72e3iT^{2} \)
67 \( 1 + (-60.0 - 60.0i)T + 4.48e3iT^{2} \)
71 \( 1 + 56.3T + 5.04e3T^{2} \)
73 \( 1 + 9.70iT - 5.32e3T^{2} \)
79 \( 1 - 84.4iT - 6.24e3T^{2} \)
83 \( 1 + (-26.7 - 26.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 115. iT - 7.92e3T^{2} \)
97 \( 1 + 146.T + 9.40e3T^{2} \)
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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

−9.939778480285957464372631251106, −9.164425149388377481143139833044, −8.228337417557633164343391548415, −7.48836699249971258584865001764, −6.75799149000026955307331881674, −5.55627145247765039340070746800, −4.82454863527788315006273666443, −3.92369011744170744693159563786, −2.55858795974231453562971438765, −1.58079296166996905378215172762, 0.19940703860348403189164376517, 1.72473336399876139875542926253, 2.91783932607166444512707845554, 3.94549696511927684795571448714, 5.20728145820055360402160887430, 5.60026034619145065570144554266, 6.86187017813296157936825561606, 7.922262573596777232253107018008, 8.108044671331008025695094921514, 9.374357743054724202122498888287

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