Properties

Label 2-276-23.18-c1-0-0
Degree $2$
Conductor $276$
Sign $-0.0252 - 0.999i$
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  + (−0.654 + 0.755i)3-s + (−0.149 + 1.03i)5-s + (0.607 + 1.33i)7-s + (−0.142 − 0.989i)9-s + (1.96 − 0.576i)11-s + (−2.86 + 6.27i)13-s + (−0.685 − 0.791i)15-s + (−6.54 + 4.20i)17-s + (5.36 + 3.44i)19-s + (−1.40 − 0.412i)21-s + (−0.692 − 4.74i)23-s + (3.74 + 1.09i)25-s + (0.841 + 0.540i)27-s + (3.48 − 2.24i)29-s + (−0.595 − 0.687i)31-s + ⋯
L(s)  = 1  + (−0.378 + 0.436i)3-s + (−0.0666 + 0.463i)5-s + (0.229 + 0.503i)7-s + (−0.0474 − 0.329i)9-s + (0.592 − 0.173i)11-s + (−0.794 + 1.73i)13-s + (−0.177 − 0.204i)15-s + (−1.58 + 1.01i)17-s + (1.22 + 0.790i)19-s + (−0.306 − 0.0899i)21-s + (−0.144 − 0.989i)23-s + (0.749 + 0.219i)25-s + (0.161 + 0.104i)27-s + (0.647 − 0.416i)29-s + (−0.107 − 0.123i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0252 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0252 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.0252 - 0.999i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ -0.0252 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.726129 + 0.744716i\)
\(L(\frac12)\) \(\approx\) \(0.726129 + 0.744716i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.654 - 0.755i)T \)
23 \( 1 + (0.692 + 4.74i)T \)
good5 \( 1 + (0.149 - 1.03i)T + (-4.79 - 1.40i)T^{2} \)
7 \( 1 + (-0.607 - 1.33i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (-1.96 + 0.576i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (2.86 - 6.27i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (6.54 - 4.20i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (-5.36 - 3.44i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (-3.48 + 2.24i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (0.595 + 0.687i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (0.580 + 4.03i)T + (-35.5 + 10.4i)T^{2} \)
41 \( 1 + (0.696 - 4.84i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-2.29 + 2.64i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + 1.92T + 47T^{2} \)
53 \( 1 + (3.25 + 7.12i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (-6.00 + 13.1i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (-4.93 - 5.69i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (-9.68 - 2.84i)T + (56.3 + 36.2i)T^{2} \)
71 \( 1 + (-0.189 - 0.0556i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (6.69 + 4.30i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (1.92 - 4.20i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (1.17 + 8.19i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (-0.180 + 0.208i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (-1.04 + 7.28i)T + (-93.0 - 27.3i)T^{2} \)
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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

−11.80149235106008814452158640350, −11.37041024119163141877566173363, −10.27035221081742800803207335628, −9.298424868296981192673446447540, −8.488508714217886982196282064059, −6.95423279634171771569431787596, −6.27711233636354653623002016821, −4.87157834262963983447643926780, −3.87866719727865197673150180352, −2.14676185148405987104574186500, 0.870798466910479748340570422877, 2.85174290760069950070485550037, 4.62236416190068586202864278485, 5.41148060714143467577820071002, 6.90597358072132300542646142072, 7.54763203140400598063970521751, 8.751202296600978466559701780129, 9.766498839585271824920221848650, 10.85046237026218807487025679079, 11.68469817590568120390359409190

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