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Properties

Label 2-1176-21.17-c1-0-7
Degree $2$
Conductor $1176$
Sign $-0.657 - 0.753i$
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  + (1.21 + 1.22i)3-s + (1.40 + 2.43i)5-s + (−0.0238 + 2.99i)9-s + (−4.74 − 2.74i)11-s + 1.35i·13-s + (−1.27 + 4.69i)15-s + (−2.88 + 5.00i)17-s + (−1.71 + 0.992i)19-s + (2.09 − 1.21i)23-s + (−1.44 + 2.49i)25-s + (−3.71 + 3.63i)27-s + 7.05i·29-s + (3.07 + 1.77i)31-s + (−2.42 − 9.17i)33-s + (−2.14 − 3.71i)37-s + ⋯
L(s)  = 1  + (0.704 + 0.709i)3-s + (0.627 + 1.08i)5-s + (−0.00795 + 0.999i)9-s + (−1.43 − 0.826i)11-s + 0.376i·13-s + (−0.329 + 1.21i)15-s + (−0.700 + 1.21i)17-s + (−0.394 + 0.227i)19-s + (0.437 − 0.252i)23-s + (−0.288 + 0.499i)25-s + (−0.715 + 0.698i)27-s + 1.31i·29-s + (0.552 + 0.318i)31-s + (−0.421 − 1.59i)33-s + (−0.352 − 0.610i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.657 - 0.753i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ -0.657 - 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.840028533\)
\(L(\frac12)\) \(\approx\) \(1.840028533\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.21 - 1.22i)T \)
7 \( 1 \)
good5 \( 1 + (-1.40 - 2.43i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.74 + 2.74i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.35iT - 13T^{2} \)
17 \( 1 + (2.88 - 5.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.71 - 0.992i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.09 + 1.21i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.05iT - 29T^{2} \)
31 \( 1 + (-3.07 - 1.77i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.14 + 3.71i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.81T + 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + (-0.201 - 0.348i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.28 + 3.04i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.28 - 2.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.75 + 2.74i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.45 + 5.97i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.08iT - 71T^{2} \)
73 \( 1 + (-0.295 - 0.170i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.19 - 2.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 + (-0.576 - 0.998i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.0iT - 97T^{2} \)
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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

−10.28288544928635519327201665193, −9.237641632863798554075128501251, −8.516808554721606030447549360819, −7.75983228816990495171109483525, −6.72046541402055453059067424130, −5.86390658378431129583465082314, −4.92130751032709253203388249977, −3.75648206149573387416653455402, −2.85458997484368416011995696917, −2.11467036155101370779813030981, 0.68530991356814670011390808408, 2.11538360085167988592118157739, 2.78627172595314431212720566453, 4.40758928183395354801021250825, 5.15312567596305895203998524518, 6.10991462725316584737970513434, 7.21555844699208934476208097470, 7.84553320233737512909825501658, 8.657027302984416406562811353284, 9.409080550043918675865190163717

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