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Properties

Label 2-1176-21.20-c1-0-0
Degree $2$
Conductor $1176$
Sign $-0.209 - 0.977i$
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  + (−0.0935 − 1.72i)3-s − 3.34·5-s + (−2.98 + 0.323i)9-s − 3.53i·11-s − 5.57i·13-s + (0.312 + 5.78i)15-s − 0.803·17-s − 0.615i·19-s + 8.47i·23-s + 6.17·25-s + (0.838 + 5.12i)27-s + 7.09i·29-s + 3.25i·31-s + (−6.10 + 0.330i)33-s + 7.31·37-s + ⋯
L(s)  = 1  + (−0.0540 − 0.998i)3-s − 1.49·5-s + (−0.994 + 0.107i)9-s − 1.06i·11-s − 1.54i·13-s + (0.0807 + 1.49i)15-s − 0.194·17-s − 0.141i·19-s + 1.76i·23-s + 1.23·25-s + (0.161 + 0.986i)27-s + 1.31i·29-s + 0.584i·31-s + (−1.06 + 0.0575i)33-s + 1.20·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.209 - 0.977i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ -0.209 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.02447361472\)
\(L(\frac12)\) \(\approx\) \(0.02447361472\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0935 + 1.72i)T \)
7 \( 1 \)
good5 \( 1 + 3.34T + 5T^{2} \)
11 \( 1 + 3.53iT - 11T^{2} \)
13 \( 1 + 5.57iT - 13T^{2} \)
17 \( 1 + 0.803T + 17T^{2} \)
19 \( 1 + 0.615iT - 19T^{2} \)
23 \( 1 - 8.47iT - 23T^{2} \)
29 \( 1 - 7.09iT - 29T^{2} \)
31 \( 1 - 3.25iT - 31T^{2} \)
37 \( 1 - 7.31T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + 6.16T + 43T^{2} \)
47 \( 1 + 7.19T + 47T^{2} \)
53 \( 1 - 2.45iT - 53T^{2} \)
59 \( 1 - 8.40T + 59T^{2} \)
61 \( 1 - 2.68iT - 61T^{2} \)
67 \( 1 + 4.00T + 67T^{2} \)
71 \( 1 + 6.75iT - 71T^{2} \)
73 \( 1 - 10.9iT - 73T^{2} \)
79 \( 1 + 9.97T + 79T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 - 9.76T + 89T^{2} \)
97 \( 1 + 13.7iT - 97T^{2} \)
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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

−10.14079530599958839625096996462, −8.720889885981034110560715312658, −8.272274411958065270594352481875, −7.58809616948480166540631552088, −6.93321804845230275013031122344, −5.81084339647323363851852898943, −5.04229239306340526507327189187, −3.47466052022910432135595560324, −3.09495818703251792206445063773, −1.21638619127873157898236953523, 0.01173520026271125750364017723, 2.29622438085860213095388388761, 3.62767401677175862224529644787, 4.44313104804493828609602568135, 4.69444754995506294305048746092, 6.29347360285304998106763517914, 7.05375027066765419894029432969, 8.099836386698885093148234409796, 8.662967942500558684518795714896, 9.650662127554368409781432197854

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