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Properties

Label 2-1152-16.13-c1-0-11
Degree $2$
Conductor $1152$
Sign $0.653 + 0.757i$
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  + (−2.68 + 2.68i)5-s − 2.15i·7-s + (1.79 − 1.79i)11-s + (−1.38 − 1.38i)13-s + 0.224·17-s + (−0.158 − 0.158i)19-s + 2.82i·23-s − 9.42i·25-s + (−1.85 − 1.85i)29-s + 1.84·31-s + (5.79 + 5.79i)35-s + (3.66 − 3.66i)37-s − 5.88i·41-s + (7.75 − 7.75i)43-s + 2.82·47-s + ⋯
L(s)  = 1  + (−1.20 + 1.20i)5-s − 0.816i·7-s + (0.542 − 0.542i)11-s + (−0.383 − 0.383i)13-s + 0.0545·17-s + (−0.0364 − 0.0364i)19-s + 0.589i·23-s − 1.88i·25-s + (−0.344 − 0.344i)29-s + 0.330·31-s + (0.980 + 0.980i)35-s + (0.603 − 0.603i)37-s − 0.918i·41-s + (1.18 − 1.18i)43-s + 0.412·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.653 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.653 + 0.757i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.653 + 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.067443707\)
\(L(\frac12)\) \(\approx\) \(1.067443707\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.68 - 2.68i)T - 5iT^{2} \)
7 \( 1 + 2.15iT - 7T^{2} \)
11 \( 1 + (-1.79 + 1.79i)T - 11iT^{2} \)
13 \( 1 + (1.38 + 1.38i)T + 13iT^{2} \)
17 \( 1 - 0.224T + 17T^{2} \)
19 \( 1 + (0.158 + 0.158i)T + 19iT^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 + (1.85 + 1.85i)T + 29iT^{2} \)
31 \( 1 - 1.84T + 31T^{2} \)
37 \( 1 + (-3.66 + 3.66i)T - 37iT^{2} \)
41 \( 1 + 5.88iT - 41T^{2} \)
43 \( 1 + (-7.75 + 7.75i)T - 43iT^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (-7.51 + 7.51i)T - 53iT^{2} \)
59 \( 1 + (-4 + 4i)T - 59iT^{2} \)
61 \( 1 + (5.98 + 5.98i)T + 61iT^{2} \)
67 \( 1 + (-10.4 - 10.4i)T + 67iT^{2} \)
71 \( 1 + 4.31iT - 71T^{2} \)
73 \( 1 - 5.97iT - 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 + (10.1 + 10.1i)T + 83iT^{2} \)
89 \( 1 + 1.42iT - 89T^{2} \)
97 \( 1 + 16.3T + 97T^{2} \)
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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

−9.876670429214528349263748253459, −8.760396424208690707655137133938, −7.78313334176697361829413615738, −7.29388375744792035222449067803, −6.59308820580236139171685036918, −5.48433530454556943902813883407, −4.02259350406508291033902256383, −3.69116963238625032193250723708, −2.52470827747262946920910925984, −0.55541557451302667134499855871, 1.14346203360746750950481671300, 2.62055533450604724530897945330, 4.04243083132926585200684888900, 4.58884203238575122857012717922, 5.51424423138371197593413482805, 6.63068423489140521544508372216, 7.63611021909963571576795143418, 8.296665014888312482195553514230, 9.087011131258422296378520969661, 9.570939978232563019117681367493

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