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Properties

Label 2-276-12.11-c1-0-39
Degree $2$
Conductor $276$
Sign $-0.923 - 0.382i$
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  + (−0.725 − 1.21i)2-s + (−0.173 − 1.72i)3-s + (−0.946 + 1.76i)4-s + 1.59i·5-s + (−1.96 + 1.46i)6-s − 4.71i·7-s + (2.82 − 0.129i)8-s + (−2.93 + 0.599i)9-s + (1.93 − 1.15i)10-s − 5.82·11-s + (3.20 + 1.32i)12-s − 1.61·13-s + (−5.72 + 3.42i)14-s + (2.74 − 0.276i)15-s + (−2.20 − 3.33i)16-s − 3.60i·17-s + ⋯
L(s)  = 1  + (−0.513 − 0.858i)2-s + (−0.100 − 0.994i)3-s + (−0.473 + 0.880i)4-s + 0.711i·5-s + (−0.802 + 0.596i)6-s − 1.78i·7-s + (0.998 − 0.0456i)8-s + (−0.979 + 0.199i)9-s + (0.610 − 0.364i)10-s − 1.75·11-s + (0.923 + 0.382i)12-s − 0.448·13-s + (−1.52 + 0.914i)14-s + (0.707 − 0.0713i)15-s + (−0.551 − 0.834i)16-s − 0.873i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.923 - 0.382i$
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: \((2,\ 276,\ (\ :1/2),\ -0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.101130 + 0.508471i\)
\(L(\frac12)\) \(\approx\) \(0.101130 + 0.508471i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.725 + 1.21i)T \)
3 \( 1 + (0.173 + 1.72i)T \)
23 \( 1 + T \)
good5 \( 1 - 1.59iT - 5T^{2} \)
7 \( 1 + 4.71iT - 7T^{2} \)
11 \( 1 + 5.82T + 11T^{2} \)
13 \( 1 + 1.61T + 13T^{2} \)
17 \( 1 + 3.60iT - 17T^{2} \)
19 \( 1 - 3.18iT - 19T^{2} \)
29 \( 1 - 0.123iT - 29T^{2} \)
31 \( 1 + 2.32iT - 31T^{2} \)
37 \( 1 - 1.69T + 37T^{2} \)
41 \( 1 + 8.92iT - 41T^{2} \)
43 \( 1 + 3.88iT - 43T^{2} \)
47 \( 1 + 4.75T + 47T^{2} \)
53 \( 1 + 8.35iT - 53T^{2} \)
59 \( 1 - 7.00T + 59T^{2} \)
61 \( 1 - 2.05T + 61T^{2} \)
67 \( 1 + 11.0iT - 67T^{2} \)
71 \( 1 + 1.48T + 71T^{2} \)
73 \( 1 - 7.31T + 73T^{2} \)
79 \( 1 - 11.3iT - 79T^{2} \)
83 \( 1 - 16.1T + 83T^{2} \)
89 \( 1 + 10.1iT - 89T^{2} \)
97 \( 1 - 2.76T + 97T^{2} \)
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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

−11.10950939928738206496641357665, −10.59026142595689246675240709069, −9.833221403662769558936640901573, −8.168228795580259165530257125749, −7.51736246160829911167795161123, −6.90828994310623514425591289855, −5.06086139337051871343685164409, −3.47126598389534929328437903695, −2.29947214924461399783900101826, −0.45059290959515320755658601361, 2.61104767887065526249458041729, 4.80184906886862181246995557522, 5.28233920800409565914121207455, 6.18223804441807025337598597367, 7.997215067110143294438650733584, 8.610166990252344465270742749975, 9.369207295152072454400568578721, 10.23778085752836920451902640975, 11.23073695261312910284088798236, 12.46857205654009119711163223636

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