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Properties

Label 2-1176-1176.485-c0-0-1
Degree $2$
Conductor $1176$
Sign $0.910 - 0.414i$
Analytic cond. $0.586900$
Root an. cond. $0.766094$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.988 + 0.149i)2-s + (0.826 + 0.563i)3-s + (0.955 − 0.294i)4-s + (0.0546 − 0.728i)5-s + (−0.900 − 0.433i)6-s + (0.0747 + 0.997i)7-s + (−0.900 + 0.433i)8-s + (0.365 + 0.930i)9-s + (0.0546 + 0.728i)10-s + (0.698 − 1.77i)11-s + (0.955 + 0.294i)12-s + (−0.222 − 0.974i)14-s + (0.455 − 0.571i)15-s + (0.826 − 0.563i)16-s + (−0.5 − 0.866i)18-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)2-s + (0.826 + 0.563i)3-s + (0.955 − 0.294i)4-s + (0.0546 − 0.728i)5-s + (−0.900 − 0.433i)6-s + (0.0747 + 0.997i)7-s + (−0.900 + 0.433i)8-s + (0.365 + 0.930i)9-s + (0.0546 + 0.728i)10-s + (0.698 − 1.77i)11-s + (0.955 + 0.294i)12-s + (−0.222 − 0.974i)14-s + (0.455 − 0.571i)15-s + (0.826 − 0.563i)16-s + (−0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.910 - 0.414i$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ 0.910 - 0.414i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9683534375\)
\(L(\frac12)\) \(\approx\) \(0.9683534375\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.988 - 0.149i)T \)
3 \( 1 + (-0.826 - 0.563i)T \)
7 \( 1 + (-0.0747 - 0.997i)T \)
good5 \( 1 + (-0.0546 + 0.728i)T + (-0.988 - 0.149i)T^{2} \)
11 \( 1 + (-0.698 + 1.77i)T + (-0.733 - 0.680i)T^{2} \)
13 \( 1 + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.0747 + 0.997i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.0747 - 0.997i)T^{2} \)
29 \( 1 + (-0.440 - 1.92i)T + (-0.900 + 0.433i)T^{2} \)
31 \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.826 - 0.563i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (-0.955 + 0.294i)T^{2} \)
53 \( 1 + (0.955 - 0.294i)T + (0.826 - 0.563i)T^{2} \)
59 \( 1 + (0.0747 + 0.997i)T + (-0.988 + 0.149i)T^{2} \)
61 \( 1 + (-0.826 - 0.563i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (-1.78 - 0.268i)T + (0.955 + 0.294i)T^{2} \)
79 \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (1.23 - 1.54i)T + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.733 - 0.680i)T^{2} \)
97 \( 1 + 1.46T + T^{2} \)
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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

−9.483306886835769844063494480825, −9.226867822349074883461046295587, −8.432105585183995343951010796352, −8.180403814989975058810996424706, −6.84991313679585504211305531683, −5.81044003988202671609047227886, −5.10260964219922024590699845952, −3.60039343042779205103150252673, −2.72020108462794129341408914602, −1.42354524659909627480525718237, 1.41214657036972096028073622368, 2.36228742016987469644531787857, 3.45176601302977470633401733657, 4.42437453597035390779312860381, 6.36147150832160744696827544439, 6.97670793633103289412384817351, 7.39911130016569918511153355201, 8.194769086382088637647489659125, 9.204166625706243575758640414402, 9.870334282933525249594777068835

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