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Properties

Label 2-1176-3.2-c2-0-48
Degree $2$
Conductor $1176$
Sign $0.797 + 0.603i$
Analytic cond. $32.0436$
Root an. cond. $5.66071$
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  + (1.80 − 2.39i)3-s + 2.81i·5-s + (−2.45 − 8.65i)9-s + 7.08i·11-s − 8.30·13-s + (6.73 + 5.08i)15-s + 1.76i·17-s + 27.3·19-s + 1.97i·23-s + 17.0·25-s + (−25.1 − 9.79i)27-s − 22.7i·29-s + 31.4·31-s + (16.9 + 12.8i)33-s + 49.5·37-s + ⋯
L(s)  = 1  + (0.603 − 0.797i)3-s + 0.562i·5-s + (−0.272 − 0.962i)9-s + 0.644i·11-s − 0.639·13-s + (0.448 + 0.339i)15-s + 0.103i·17-s + 1.44·19-s + 0.0859i·23-s + 0.683·25-s + (−0.931 − 0.362i)27-s − 0.782i·29-s + 1.01·31-s + (0.513 + 0.388i)33-s + 1.33·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 + 0.603i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.797 + 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.797 + 0.603i$
Analytic conductor: \(32.0436\)
Root analytic conductor: \(5.66071\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (785, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1),\ 0.797 + 0.603i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.429827426\)
\(L(\frac12)\) \(\approx\) \(2.429827426\)
\(L(2)\) 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 \( 1 + (-1.80 + 2.39i)T \)
7 \( 1 \)
good5 \( 1 - 2.81iT - 25T^{2} \)
11 \( 1 - 7.08iT - 121T^{2} \)
13 \( 1 + 8.30T + 169T^{2} \)
17 \( 1 - 1.76iT - 289T^{2} \)
19 \( 1 - 27.3T + 361T^{2} \)
23 \( 1 - 1.97iT - 529T^{2} \)
29 \( 1 + 22.7iT - 841T^{2} \)
31 \( 1 - 31.4T + 961T^{2} \)
37 \( 1 - 49.5T + 1.36e3T^{2} \)
41 \( 1 + 34.3iT - 1.68e3T^{2} \)
43 \( 1 - 30.0T + 1.84e3T^{2} \)
47 \( 1 + 71.4iT - 2.20e3T^{2} \)
53 \( 1 - 52.7iT - 2.80e3T^{2} \)
59 \( 1 + 36.7iT - 3.48e3T^{2} \)
61 \( 1 - 63.8T + 3.72e3T^{2} \)
67 \( 1 + 81.7T + 4.48e3T^{2} \)
71 \( 1 - 124. iT - 5.04e3T^{2} \)
73 \( 1 - 97.3T + 5.32e3T^{2} \)
79 \( 1 - 15.7T + 6.24e3T^{2} \)
83 \( 1 - 86.0iT - 6.88e3T^{2} \)
89 \( 1 + 71.8iT - 7.92e3T^{2} \)
97 \( 1 - 162.T + 9.40e3T^{2} \)
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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

−9.567385708877229409968834625022, −8.588324107599531390861116170108, −7.65352113604068414592532009352, −7.18845139029948933400805919388, −6.37954659912369130103364617894, −5.33436319705115148946064825757, −4.11108095230880140585073965020, −2.97700639159047140978374113436, −2.24944417444379927989548021474, −0.869025546660470156716314802708, 1.00380746940348002798633909448, 2.61835057413244604628876790654, 3.39544167842275440794339653422, 4.59869929459273232190615904671, 5.12410440168447595182748836414, 6.18418257147328553430709170203, 7.48704539329495186575517221023, 8.103811059698569246225126891200, 9.026163031537234494577315032925, 9.503069348328937585138859639984

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