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Properties

Label 2-7200-12.11-c1-0-8
Degree $2$
Conductor $7200$
Sign $-0.985 + 0.169i$
Analytic cond. $57.4922$
Root an. cond. $7.58236$
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  + 3.50i·7-s + 1.92·11-s − 5.50·13-s + 4.44i·17-s + 7.00i·19-s + 1.10·23-s + 5.47i·29-s + 8.28i·31-s + 0.778·37-s − 2.44i·41-s − 9.55i·43-s − 11.7·47-s − 5.28·49-s − 11.5i·53-s + 9.78·59-s + ⋯
L(s)  = 1  + 1.32i·7-s + 0.581·11-s − 1.52·13-s + 1.07i·17-s + 1.60i·19-s + 0.229·23-s + 1.01i·29-s + 1.48i·31-s + 0.127·37-s − 0.381i·41-s − 1.45i·43-s − 1.70·47-s − 0.754·49-s − 1.58i·53-s + 1.27·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.985 + 0.169i$
Analytic conductor: \(57.4922\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7200} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ -0.985 + 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8422427247\)
\(L(\frac12)\) \(\approx\) \(0.8422427247\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 - 3.50iT - 7T^{2} \)
11 \( 1 - 1.92T + 11T^{2} \)
13 \( 1 + 5.50T + 13T^{2} \)
17 \( 1 - 4.44iT - 17T^{2} \)
19 \( 1 - 7.00iT - 19T^{2} \)
23 \( 1 - 1.10T + 23T^{2} \)
29 \( 1 - 5.47iT - 29T^{2} \)
31 \( 1 - 8.28iT - 31T^{2} \)
37 \( 1 - 0.778T + 37T^{2} \)
41 \( 1 + 2.44iT - 41T^{2} \)
43 \( 1 + 9.55iT - 43T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 + 11.5iT - 53T^{2} \)
59 \( 1 - 9.78T + 59T^{2} \)
61 \( 1 + 3.45T + 61T^{2} \)
67 \( 1 + 5.45iT - 67T^{2} \)
71 \( 1 - 4.25T + 71T^{2} \)
73 \( 1 + 7.27T + 73T^{2} \)
79 \( 1 + 2.82iT - 79T^{2} \)
83 \( 1 - 4.25T + 83T^{2} \)
89 \( 1 + 0.386iT - 89T^{2} \)
97 \( 1 - 9.29T + 97T^{2} \)
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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

−8.459957426627727618702384424770, −7.63136748700994967508645914457, −6.81240310881956721827772306098, −6.22526830766590838155651244676, −5.33033206531766543723794171215, −5.03683413648341629996790428109, −3.85102603343282751981431363188, −3.20178722309666073594148083722, −2.15943548609931364284889718006, −1.57768942228682337509732065066, 0.21620680389475626313847420385, 1.04012547736529271674059685524, 2.39869727933498984934269253484, 3.00110292755872942563450613516, 4.20161472166326309883460648270, 4.54155848373246237791831296814, 5.27407848782790239125848448276, 6.39835109324285259690595000807, 6.93368045830600577026987242351, 7.51983184183217498351114351621

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