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Properties

Label 2-1176-168.83-c0-0-7
Degree $2$
Conductor $1176$
Sign $-0.999 + 0.0287i$
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  − i·2-s + (0.382 − 0.923i)3-s − 4-s + (−0.923 − 0.382i)6-s + i·8-s + (−0.707 − 0.707i)9-s − 1.41i·11-s + (−0.382 + 0.923i)12-s + 16-s − 1.84·17-s + (−0.707 + 0.707i)18-s − 0.765i·19-s − 1.41·22-s + (0.923 + 0.382i)24-s + 25-s + ⋯
L(s)  = 1  − i·2-s + (0.382 − 0.923i)3-s − 4-s + (−0.923 − 0.382i)6-s + i·8-s + (−0.707 − 0.707i)9-s − 1.41i·11-s + (−0.382 + 0.923i)12-s + 16-s − 1.84·17-s + (−0.707 + 0.707i)18-s − 0.765i·19-s − 1.41·22-s + (0.923 + 0.382i)24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.999 + 0.0287i$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ -0.999 + 0.0287i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9124818595\)
\(L(\frac12)\) \(\approx\) \(0.9124818595\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.382 + 0.923i)T \)
7 \( 1 \)
good5 \( 1 - T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + 1.84T + T^{2} \)
19 \( 1 + 0.765iT - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 0.765T + T^{2} \)
43 \( 1 - 1.41T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 1.84T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 1.84iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.84T + T^{2} \)
89 \( 1 - 0.765T + T^{2} \)
97 \( 1 + 0.765iT - T^{2} \)
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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

−9.373643276338087869656955156343, −8.725269070750631738736679442146, −8.337365684128832299456174453018, −7.10437171159190464460889945908, −6.25936685919341133728656368833, −5.22163347925512963194270508568, −4.04191209190160151007295168392, −2.99336505608760436737192229652, −2.23247308235490080909180809778, −0.78489882368606348339151638485, 2.25014861699627665748158286662, 3.72211330156473962760712065813, 4.53614274509186421039274329696, 5.08851632627921335379438117639, 6.27135581390868529606822408346, 7.09327806826927515740467569862, 7.947057195835617272580580665273, 8.844312514439436965730854805893, 9.315580957594600660583127132615, 10.17749797607537853875596788158

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