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Properties

Label 2-1176-7.2-c1-0-16
Degree $2$
Conductor $1176$
Sign $0.386 + 0.922i$
Analytic cond. $9.39040$
Root an. cond. $3.06437$
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  + (0.5 + 0.866i)3-s + (1 − 1.73i)5-s + (−0.499 + 0.866i)9-s + (−2 − 3.46i)11-s − 2·13-s + 1.99·15-s + (−1 − 1.73i)17-s + (2 − 3.46i)19-s + (4 − 6.92i)23-s + (0.500 + 0.866i)25-s − 0.999·27-s + 6·29-s + (−4 − 6.92i)31-s + (1.99 − 3.46i)33-s + (−3 + 5.19i)37-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.447 − 0.774i)5-s + (−0.166 + 0.288i)9-s + (−0.603 − 1.04i)11-s − 0.554·13-s + 0.516·15-s + (−0.242 − 0.420i)17-s + (0.458 − 0.794i)19-s + (0.834 − 1.44i)23-s + (0.100 + 0.173i)25-s − 0.192·27-s + 1.11·29-s + (−0.718 − 1.24i)31-s + (0.348 − 0.603i)33-s + (−0.493 + 0.854i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.630172600\)
\(L(\frac12)\) \(\approx\) \(1.630172600\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−9.483509639300566864530743596231, −8.859029203162157645556683985493, −8.253465339169081962557179281522, −7.19606435561194585456817772159, −6.14364745377578558559126635252, −5.08427308143170178175268646645, −4.71275716493911233208348817303, −3.27481007839388137981136165915, −2.39971869454034995860714342133, −0.67598066245933542679691704562, 1.62060026121176131173092243339, 2.58325468363444107765005096270, 3.53850914002634254838330623309, 4.90602183779018520341297399732, 5.75164118249395832507971187875, 6.90671324548128995301522749854, 7.23974721291906822564795327381, 8.180636641101783764424918200312, 9.166535085941774680575927494016, 10.02426693125771851321324607233

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