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Properties

Label 2-276-12.11-c1-0-25
Degree $2$
Conductor $276$
Sign $0.431 - 0.901i$
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  + (1.15 + 0.809i)2-s + (1.20 + 1.24i)3-s + (0.689 + 1.87i)4-s − 3.47i·5-s + (0.397 + 2.41i)6-s + 0.968i·7-s + (−0.720 + 2.73i)8-s + (−0.0786 + 2.99i)9-s + (2.81 − 4.03i)10-s + 2.17·11-s + (−1.49 + 3.12i)12-s − 3.21·13-s + (−0.783 + 1.12i)14-s + (4.31 − 4.20i)15-s + (−3.04 + 2.58i)16-s − 1.26i·17-s + ⋯
L(s)  = 1  + (0.819 + 0.572i)2-s + (0.697 + 0.716i)3-s + (0.344 + 0.938i)4-s − 1.55i·5-s + (0.162 + 0.986i)6-s + 0.366i·7-s + (−0.254 + 0.967i)8-s + (−0.0262 + 0.999i)9-s + (0.890 − 1.27i)10-s + 0.655·11-s + (−0.431 + 0.901i)12-s − 0.890·13-s + (−0.209 + 0.300i)14-s + (1.11 − 1.08i)15-s + (−0.762 + 0.647i)16-s − 0.306i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.431 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.431 - 0.901i$
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: \((2,\ 276,\ (\ :1/2),\ 0.431 - 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97822 + 1.24605i\)
\(L(\frac12)\) \(\approx\) \(1.97822 + 1.24605i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 - 0.809i)T \)
3 \( 1 + (-1.20 - 1.24i)T \)
23 \( 1 + T \)
good5 \( 1 + 3.47iT - 5T^{2} \)
7 \( 1 - 0.968iT - 7T^{2} \)
11 \( 1 - 2.17T + 11T^{2} \)
13 \( 1 + 3.21T + 13T^{2} \)
17 \( 1 + 1.26iT - 17T^{2} \)
19 \( 1 + 4.03iT - 19T^{2} \)
29 \( 1 + 8.56iT - 29T^{2} \)
31 \( 1 + 1.32iT - 31T^{2} \)
37 \( 1 + 10.0T + 37T^{2} \)
41 \( 1 - 7.60iT - 41T^{2} \)
43 \( 1 - 1.64iT - 43T^{2} \)
47 \( 1 - 12.6T + 47T^{2} \)
53 \( 1 + 8.41iT - 53T^{2} \)
59 \( 1 - 1.24T + 59T^{2} \)
61 \( 1 - 5.28T + 61T^{2} \)
67 \( 1 - 14.1iT - 67T^{2} \)
71 \( 1 + 12.0T + 71T^{2} \)
73 \( 1 - 4.42T + 73T^{2} \)
79 \( 1 - 7.01iT - 79T^{2} \)
83 \( 1 + 8.36T + 83T^{2} \)
89 \( 1 + 3.00iT - 89T^{2} \)
97 \( 1 + 9.68T + 97T^{2} \)
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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

−12.20926023375454284885135820039, −11.50067671899924514660116418435, −9.829497220987527012013475853185, −8.956745254089588560502637083636, −8.352776092397028518514248038163, −7.23312909834527609968931323053, −5.62974253776119115780695641102, −4.78890508253787271680347222883, −4.03848703375849600837137173706, −2.43033761883011577192007742323, 1.88245988167438580809703628200, 3.08323354406607068844251902789, 3.91571623511037280122311149745, 5.79493815123384266266715509142, 6.92009681783072305297544626984, 7.30313912177562804834062668319, 8.982446381683876081608915290018, 10.19868256977775187737051308512, 10.74814087117529464979653719851, 12.02941574962461303763815913341

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