Properties

Label 2-276-276.95-c1-0-15
Degree $2$
Conductor $276$
Sign $-0.0446 + 0.999i$
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  + (−1.09 − 0.898i)2-s + (−1.50 − 0.856i)3-s + (0.384 + 1.96i)4-s + (0.204 + 0.177i)5-s + (0.873 + 2.28i)6-s + (0.657 + 1.02i)7-s + (1.34 − 2.48i)8-s + (1.53 + 2.57i)9-s + (−0.0641 − 0.377i)10-s + (0.788 − 5.48i)11-s + (1.10 − 3.28i)12-s + (3.04 + 1.95i)13-s + (0.201 − 1.70i)14-s + (−0.156 − 0.442i)15-s + (−3.70 + 1.50i)16-s + (0.519 − 1.76i)17-s + ⋯
L(s)  = 1  + (−0.772 − 0.635i)2-s + (−0.869 − 0.494i)3-s + (0.192 + 0.981i)4-s + (0.0916 + 0.0793i)5-s + (0.356 + 0.934i)6-s + (0.248 + 0.386i)7-s + (0.475 − 0.879i)8-s + (0.510 + 0.859i)9-s + (−0.0202 − 0.119i)10-s + (0.237 − 1.65i)11-s + (0.318 − 0.947i)12-s + (0.844 + 0.543i)13-s + (0.0538 − 0.456i)14-s + (−0.0403 − 0.114i)15-s + (−0.926 + 0.377i)16-s + (0.126 − 0.429i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0446 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0446 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.0446 + 0.999i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ -0.0446 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.475040 - 0.496741i\)
\(L(\frac12)\) \(\approx\) \(0.475040 - 0.496741i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.09 + 0.898i)T \)
3 \( 1 + (1.50 + 0.856i)T \)
23 \( 1 + (1.85 - 4.42i)T \)
good5 \( 1 + (-0.204 - 0.177i)T + (0.711 + 4.94i)T^{2} \)
7 \( 1 + (-0.657 - 1.02i)T + (-2.90 + 6.36i)T^{2} \)
11 \( 1 + (-0.788 + 5.48i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (-3.04 - 1.95i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (-0.519 + 1.76i)T + (-14.3 - 9.19i)T^{2} \)
19 \( 1 + (1.50 + 5.13i)T + (-15.9 + 10.2i)T^{2} \)
29 \( 1 + (-2.02 + 6.88i)T + (-24.3 - 15.6i)T^{2} \)
31 \( 1 + (-4.90 + 2.24i)T + (20.3 - 23.4i)T^{2} \)
37 \( 1 + (-0.479 - 0.552i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (1.04 + 0.901i)T + (5.83 + 40.5i)T^{2} \)
43 \( 1 + (-9.19 - 4.19i)T + (28.1 + 32.4i)T^{2} \)
47 \( 1 + 4.82T + 47T^{2} \)
53 \( 1 + (-3.63 - 5.65i)T + (-22.0 + 48.2i)T^{2} \)
59 \( 1 + (-7.03 - 4.52i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (1.89 + 4.14i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (4.32 - 0.621i)T + (64.2 - 18.8i)T^{2} \)
71 \( 1 + (-1.22 - 8.48i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (-9.52 + 2.79i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (5.07 - 7.89i)T + (-32.8 - 71.8i)T^{2} \)
83 \( 1 + (0.584 + 0.674i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (6.78 + 3.10i)T + (58.2 + 67.2i)T^{2} \)
97 \( 1 + (4.59 - 5.30i)T + (-13.8 - 96.0i)T^{2} \)
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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

−11.40688869477995164099981984642, −11.07088627498461633355091842276, −9.869862917601338999834896981239, −8.728719354309895066555517644277, −7.982958927817673976971878527339, −6.67481901321689257477017829709, −5.84227776530180362471138363294, −4.19734483569006508101505182759, −2.53799111421997646913281624794, −0.862577548520505987409649444816, 1.44938201940749478256186125394, 4.12950758557272694427570522471, 5.20845605178256062660820349573, 6.25955654044017317567107457114, 7.14739806691331186516061011313, 8.266096357051215411258383782014, 9.405754522239276782295747142642, 10.33456018300487171900307277694, 10.70464703046519034114482171504, 11.99122206516528512549913844986

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