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Properties

Label 2-276-12.11-c1-0-3
Degree $2$
Conductor $276$
Sign $-0.109 - 0.993i$
Analytic cond. $2.20387$
Root an. cond. $1.48454$
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  + (−1.40 + 0.146i)2-s + (0.169 − 1.72i)3-s + (1.95 − 0.413i)4-s + 2.76i·5-s + (0.0141 + 2.44i)6-s + 3.82i·7-s + (−2.69 + 0.868i)8-s + (−2.94 − 0.585i)9-s + (−0.406 − 3.89i)10-s − 2.92·11-s + (−0.379 − 3.44i)12-s − 4.70·13-s + (−0.561 − 5.38i)14-s + (4.76 + 0.469i)15-s + (3.65 − 1.61i)16-s + 4.06i·17-s + ⋯
L(s)  = 1  + (−0.994 + 0.103i)2-s + (0.0980 − 0.995i)3-s + (0.978 − 0.206i)4-s + 1.23i·5-s + (0.00579 + 0.999i)6-s + 1.44i·7-s + (−0.951 + 0.307i)8-s + (−0.980 − 0.195i)9-s + (−0.128 − 1.23i)10-s − 0.883·11-s + (−0.109 − 0.993i)12-s − 1.30·13-s + (−0.150 − 1.43i)14-s + (1.23 + 0.121i)15-s + (0.914 − 0.404i)16-s + 0.985i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.109 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.109 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.109 - 0.993i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ -0.109 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.380112 + 0.424327i\)
\(L(\frac12)\) \(\approx\) \(0.380112 + 0.424327i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.146i)T \)
3 \( 1 + (-0.169 + 1.72i)T \)
23 \( 1 - T \)
good5 \( 1 - 2.76iT - 5T^{2} \)
7 \( 1 - 3.82iT - 7T^{2} \)
11 \( 1 + 2.92T + 11T^{2} \)
13 \( 1 + 4.70T + 13T^{2} \)
17 \( 1 - 4.06iT - 17T^{2} \)
19 \( 1 - 3.75iT - 19T^{2} \)
29 \( 1 + 8.01iT - 29T^{2} \)
31 \( 1 - 1.15iT - 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 + 1.54iT - 41T^{2} \)
43 \( 1 - 2.22iT - 43T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 - 6.30iT - 53T^{2} \)
59 \( 1 + 7.03T + 59T^{2} \)
61 \( 1 + 6.15T + 61T^{2} \)
67 \( 1 + 1.65iT - 67T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 + 5.68T + 73T^{2} \)
79 \( 1 + 3.06iT - 79T^{2} \)
83 \( 1 + 2.88T + 83T^{2} \)
89 \( 1 - 8.17iT - 89T^{2} \)
97 \( 1 + 4.28T + 97T^{2} \)
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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

−12.07705317860618361565907451476, −11.15283494318419760497985777146, −10.22090282496838923037866739387, −9.203041898647918150314702256618, −8.030263495155313903085055538212, −7.53156843267487049040046113421, −6.31866829099666055748011104551, −5.71475185023637923914633719201, −2.77794448176161781270699095796, −2.25792569689683547052017900177, 0.55057673232534083751043900377, 2.81248625928597126450061153501, 4.43786046395680433575998954721, 5.26428810609676874017099043362, 7.12016626763835983709706284179, 7.905352335645394724190708940460, 9.035285222146517074832551548123, 9.654840461087711188864296333123, 10.49911024221925137132468376245, 11.24712680038896876840728260326

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