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Properties

Label 2-276-92.91-c1-0-18
Degree $2$
Conductor $276$
Sign $0.111 + 0.993i$
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  + (1.11 − 0.872i)2-s − i·3-s + (0.477 − 1.94i)4-s + 0.970i·5-s + (−0.872 − 1.11i)6-s + 4.31·7-s + (−1.16 − 2.57i)8-s − 9-s + (0.846 + 1.07i)10-s − 3.90·11-s + (−1.94 − 0.477i)12-s + 1.84·13-s + (4.80 − 3.76i)14-s + 0.970·15-s + (−3.54 − 1.85i)16-s + 0.465i·17-s + ⋯
L(s)  = 1  + (0.786 − 0.617i)2-s − 0.577i·3-s + (0.238 − 0.971i)4-s + 0.433i·5-s + (−0.356 − 0.454i)6-s + 1.63·7-s + (−0.411 − 0.911i)8-s − 0.333·9-s + (0.267 + 0.341i)10-s − 1.17·11-s + (−0.560 − 0.137i)12-s + 0.511·13-s + (1.28 − 1.00i)14-s + 0.250·15-s + (−0.886 − 0.463i)16-s + 0.112i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.111 + 0.993i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ 0.111 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51828 - 1.35795i\)
\(L(\frac12)\) \(\approx\) \(1.51828 - 1.35795i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.11 + 0.872i)T \)
3 \( 1 + iT \)
23 \( 1 + (-1.65 - 4.50i)T \)
good5 \( 1 - 0.970iT - 5T^{2} \)
7 \( 1 - 4.31T + 7T^{2} \)
11 \( 1 + 3.90T + 11T^{2} \)
13 \( 1 - 1.84T + 13T^{2} \)
17 \( 1 - 0.465iT - 17T^{2} \)
19 \( 1 + 6.89T + 19T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 + 2.44iT - 31T^{2} \)
37 \( 1 - 11.3iT - 37T^{2} \)
41 \( 1 + 2.17T + 41T^{2} \)
43 \( 1 - 5.37T + 43T^{2} \)
47 \( 1 - 8.65iT - 47T^{2} \)
53 \( 1 + 10.2iT - 53T^{2} \)
59 \( 1 - 8.30iT - 59T^{2} \)
61 \( 1 + 7.91iT - 61T^{2} \)
67 \( 1 - 2.57T + 67T^{2} \)
71 \( 1 + 6.53iT - 71T^{2} \)
73 \( 1 + 1.48T + 73T^{2} \)
79 \( 1 + 12.2T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + 4.10iT - 89T^{2} \)
97 \( 1 + 5.32iT - 97T^{2} \)
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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

−11.52895387316905061886176041468, −11.01844204663193917847403004761, −10.27416717153068837719400380482, −8.638394610971583053040345539447, −7.74728744601061923675266699841, −6.53397896034522431729266915619, −5.39081204972518328746471208413, −4.44971312888239486939809767184, −2.80743374233648907176675589015, −1.60589556141370199252907766543, 2.42448247015928798056233635288, 4.19895853352835206624838952014, 4.89216799882310164732284975474, 5.74505674351516153882788560138, 7.22169002200395508852714111810, 8.410063704571854522705821907976, 8.658970490474626496776421678245, 10.65919113624465762109057840615, 11.04552758336897131687599799456, 12.30370659303714817533046568896

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