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Properties

Label 2-1176-24.5-c0-0-5
Degree $2$
Conductor $1176$
Sign $-0.707 + 0.707i$
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 + (−0.707 − 0.707i)3-s − 4-s + 1.41·5-s + (−0.707 + 0.707i)6-s + i·8-s + 1.00i·9-s − 1.41i·10-s + (0.707 + 0.707i)12-s − 1.41i·13-s + (−1.00 − 1.00i)15-s + 16-s + 1.00·18-s − 1.41i·19-s − 1.41·20-s + ⋯
L(s)  = 1  − i·2-s + (−0.707 − 0.707i)3-s − 4-s + 1.41·5-s + (−0.707 + 0.707i)6-s + i·8-s + 1.00i·9-s − 1.41i·10-s + (0.707 + 0.707i)12-s − 1.41i·13-s + (−1.00 − 1.00i)15-s + 16-s + 1.00·18-s − 1.41i·19-s − 1.41·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9281385406\)
\(L(\frac12)\) \(\approx\) \(0.9281385406\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 \)
good5 \( 1 - 1.41T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 - 1.41iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.915337593459472176950716195080, −9.071073559719223987239948701358, −8.168779821431642837264502349847, −7.12974871116930296976184444068, −6.01388706414189450613809590836, −5.44920073598519024218183916796, −4.66571462629625165267689382558, −2.97751277729535171112969470598, −2.18414110671397883270716281429, −0.995864069302582513758742616161, 1.69004547909217054275099765028, 3.57657345766192185641824632263, 4.58483457073773769027064236661, 5.33883496385505886649420636272, 6.24223493884579396458614411424, 6.46106084519161999254711920095, 7.71026345054088464067192211607, 8.937522173988053774085824596038, 9.405386232152460870724739136522, 10.03606291628055927229513639428

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