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Properties

Label 2-1152-16.13-c1-0-2
Degree $2$
Conductor $1152$
Sign $-0.983 + 0.179i$
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  + (−2.37 + 2.37i)5-s + 3.64i·7-s + (−0.841 + 0.841i)11-s + (2.64 + 2.64i)13-s − 3.06·17-s + (−1.64 − 1.64i)19-s − 7.82i·23-s − 6.29i·25-s + (−0.692 − 0.692i)29-s − 0.354·31-s + (−8.66 − 8.66i)35-s + (−4.64 + 4.64i)37-s + 6.43i·41-s + (5.64 − 5.64i)43-s − 11.1·47-s + ⋯
L(s)  = 1  + (−1.06 + 1.06i)5-s + 1.37i·7-s + (−0.253 + 0.253i)11-s + (0.733 + 0.733i)13-s − 0.744·17-s + (−0.377 − 0.377i)19-s − 1.63i·23-s − 1.25i·25-s + (−0.128 − 0.128i)29-s − 0.0636·31-s + (−1.46 − 1.46i)35-s + (−0.763 + 0.763i)37-s + 1.00i·41-s + (0.860 − 0.860i)43-s − 1.63·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.983 + 0.179i$
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: \((2,\ 1152,\ (\ :1/2),\ -0.983 + 0.179i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5797012665\)
\(L(\frac12)\) \(\approx\) \(0.5797012665\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.37 - 2.37i)T - 5iT^{2} \)
7 \( 1 - 3.64iT - 7T^{2} \)
11 \( 1 + (0.841 - 0.841i)T - 11iT^{2} \)
13 \( 1 + (-2.64 - 2.64i)T + 13iT^{2} \)
17 \( 1 + 3.06T + 17T^{2} \)
19 \( 1 + (1.64 + 1.64i)T + 19iT^{2} \)
23 \( 1 + 7.82iT - 23T^{2} \)
29 \( 1 + (0.692 + 0.692i)T + 29iT^{2} \)
31 \( 1 + 0.354T + 31T^{2} \)
37 \( 1 + (4.64 - 4.64i)T - 37iT^{2} \)
41 \( 1 - 6.43iT - 41T^{2} \)
43 \( 1 + (-5.64 + 5.64i)T - 43iT^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 + (5.44 - 5.44i)T - 53iT^{2} \)
59 \( 1 + (-7.82 + 7.82i)T - 59iT^{2} \)
61 \( 1 + (4.64 + 4.64i)T + 61iT^{2} \)
67 \( 1 + (4 + 4i)T + 67iT^{2} \)
71 \( 1 - 3.36iT - 71T^{2} \)
73 \( 1 - 7.29iT - 73T^{2} \)
79 \( 1 - 4.35T + 79T^{2} \)
83 \( 1 + (-0.841 - 0.841i)T + 83iT^{2} \)
89 \( 1 - 9.50iT - 89T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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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

−10.41198465987766141180538680986, −9.243515469951575873378789660189, −8.554369069857250590133500107413, −7.88199027480367285863655087840, −6.62921194369215893626846100425, −6.43846738688178523646398319049, −5.01765125613985346521737802137, −4.08845189282429991079967069240, −2.99595667346557616200554501331, −2.16259026528002634596770726423, 0.25985393226455843705158104790, 1.41457787333988374001295658614, 3.43872942457416310861972626210, 4.01006401473246068063172038379, 4.89002235280462998451095744386, 5.89394642964229270078127337513, 7.14853276863552588343487791667, 7.76310643093826980520719136375, 8.426224196565225373897484659142, 9.228829683065505588957499961004

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