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Properties

Label 2-2775-555.359-c0-0-1
Degree $2$
Conductor $2775$
Sign $0.996 + 0.0870i$
Analytic cond. $1.38490$
Root an. cond. $1.17682$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + (0.866 + 0.5i)7-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)12-s + (−0.866 − 0.5i)13-s + (−0.499 − 0.866i)16-s + (1 − 1.73i)19-s + (0.499 + 0.866i)21-s + 0.999i·27-s + (0.866 − 0.499i)28-s − 31-s + 0.999·36-s + i·37-s + (−0.499 − 0.866i)39-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + (0.866 + 0.5i)7-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)12-s + (−0.866 − 0.5i)13-s + (−0.499 − 0.866i)16-s + (1 − 1.73i)19-s + (0.499 + 0.866i)21-s + 0.999i·27-s + (0.866 − 0.499i)28-s − 31-s + 0.999·36-s + i·37-s + (−0.499 − 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2775\)    =    \(3 \cdot 5^{2} \cdot 37\)
Sign: $0.996 + 0.0870i$
Analytic conductor: \(1.38490\)
Root analytic conductor: \(1.17682\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2775} (2024, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2775,\ (\ :0),\ 0.996 + 0.0870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.003217397\)
\(L(\frac12)\) \(\approx\) \(2.003217397\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
37 \( 1 - iT \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + iT - T^{2} \)
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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

−9.072854504674134798633438857845, −8.317688603199391419565509898511, −7.43967487886012268589648347306, −6.95082059795359754537615854739, −5.63707782312064713724533046933, −5.07261599624894089950998795339, −4.47510087751680836832660732501, −3.00507688105172873945131661509, −2.46580081951396923268192326839, −1.39691385612135329514453256174, 1.58968975608278260815546717874, 2.24627510075681699788047842975, 3.42844506330129041196371243280, 3.92876626880542804710367814715, 4.99893724517073529666204536598, 6.16760697365510773660104628599, 7.08556191020086172107537252912, 7.71217654638816315425079421353, 7.87848882703527999890408898556, 8.879222794119747256597745361372

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