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Properties

Label 2-1176-21.5-c1-0-33
Degree $2$
Conductor $1176$
Sign $-0.719 + 0.694i$
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  + (−1.02 − 1.39i)3-s + (2.00 − 3.47i)5-s + (−0.900 + 2.86i)9-s + (3.26 − 1.88i)11-s − 4.48i·13-s + (−6.91 + 0.758i)15-s + (−2.05 − 3.56i)17-s + (5.69 + 3.28i)19-s + (4.33 + 2.50i)23-s + (−5.56 − 9.63i)25-s + (4.91 − 1.67i)27-s − 1.23i·29-s + (−0.809 + 0.467i)31-s + (−5.98 − 2.62i)33-s + (−1.01 + 1.75i)37-s + ⋯
L(s)  = 1  + (−0.591 − 0.806i)3-s + (0.897 − 1.55i)5-s + (−0.300 + 0.953i)9-s + (0.984 − 0.568i)11-s − 1.24i·13-s + (−1.78 + 0.195i)15-s + (−0.499 − 0.865i)17-s + (1.30 + 0.753i)19-s + (0.904 + 0.522i)23-s + (−1.11 − 1.92i)25-s + (0.946 − 0.322i)27-s − 0.228i·29-s + (−0.145 + 0.0839i)31-s + (−1.04 − 0.457i)33-s + (−0.166 + 0.288i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.719 + 0.694i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ -0.719 + 0.694i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.602152266\)
\(L(\frac12)\) \(\approx\) \(1.602152266\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.02 + 1.39i)T \)
7 \( 1 \)
good5 \( 1 + (-2.00 + 3.47i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.26 + 1.88i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.48iT - 13T^{2} \)
17 \( 1 + (2.05 + 3.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.69 - 3.28i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.33 - 2.50i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.23iT - 29T^{2} \)
31 \( 1 + (0.809 - 0.467i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.01 - 1.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.21T + 41T^{2} \)
43 \( 1 + 3.14T + 43T^{2} \)
47 \( 1 + (2.92 - 5.07i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.95 - 2.28i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.982 - 1.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.00 - 4.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.88 + 6.72i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.97iT - 71T^{2} \)
73 \( 1 + (12.8 - 7.41i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.85 - 3.20i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.15T + 83T^{2} \)
89 \( 1 + (-6.53 + 11.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.06iT - 97T^{2} \)
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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

−9.392413258064569802376148067193, −8.654467078048136769094681098207, −7.88018801084628637121986106705, −6.91427087831815227161998841342, −5.77280444537212865828132109344, −5.50130809309812186261251882560, −4.57899052820795954107474218400, −3.00050595800803128659138899581, −1.49306852088425134059360469048, −0.825292372065571275092538731223, 1.75956803848388461262100100893, 3.01834799024711530974352904602, 3.95933523764144316587290284155, 4.97052772348929431356979779258, 6.03778806469080441295861036019, 6.74382933196220916361983066563, 7.09301411697651855116039725405, 8.862056875057230318059266512778, 9.477313364647813016906180513470, 10.04016993703550196396472198065

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