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Properties

Label 2-1775-1775.354-c0-0-0
Degree $2$
Conductor $1775$
Sign $-0.994 - 0.107i$
Analytic cond. $0.885840$
Root an. cond. $0.941190$
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  + (0.170 + 0.0554i)2-s + (−0.413 + 0.568i)3-s + (−0.783 − 0.568i)4-s + (−0.473 + 0.880i)5-s + (−0.101 + 0.0740i)6-s + (−0.207 − 0.285i)8-s + (0.156 + 0.481i)9-s + (−0.129 + 0.123i)10-s + (0.646 − 0.210i)12-s + (−0.304 − 0.633i)15-s + (0.279 + 0.860i)16-s + 0.0907i·18-s + (−0.635 + 0.462i)19-s + (0.872 − 0.419i)20-s + 0.247·24-s + (−0.550 − 0.834i)25-s + ⋯
L(s)  = 1  + (0.170 + 0.0554i)2-s + (−0.413 + 0.568i)3-s + (−0.783 − 0.568i)4-s + (−0.473 + 0.880i)5-s + (−0.101 + 0.0740i)6-s + (−0.207 − 0.285i)8-s + (0.156 + 0.481i)9-s + (−0.129 + 0.123i)10-s + (0.646 − 0.210i)12-s + (−0.304 − 0.633i)15-s + (0.279 + 0.860i)16-s + 0.0907i·18-s + (−0.635 + 0.462i)19-s + (0.872 − 0.419i)20-s + 0.247·24-s + (−0.550 − 0.834i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1775\)    =    \(5^{2} \cdot 71\)
Sign: $-0.994 - 0.107i$
Analytic conductor: \(0.885840\)
Root analytic conductor: \(0.941190\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1775} (354, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1775,\ (\ :0),\ -0.994 - 0.107i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3386709397\)
\(L(\frac12)\) \(\approx\) \(0.3386709397\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.473 - 0.880i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (-0.170 - 0.0554i)T + (0.809 + 0.587i)T^{2} \)
3 \( 1 + (0.413 - 0.568i)T + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.635 - 0.462i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (1.55 + 1.13i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (1.74 - 0.568i)T + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - 1.98iT - T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (1.88 + 0.612i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-1.59 - 1.15i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (1.14 + 1.57i)T + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.578 + 1.78i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.309 + 0.951i)T^{2} \)
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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

−10.03975976347451504913201940872, −9.269424904997242709110413119270, −8.236173824955913800521482035894, −7.56281879184591362198075496463, −6.45839416534949939412283567903, −5.80405143635179928753294753927, −4.87676758975775058212818183460, −4.18240656794258010819114369932, −3.38518019701708921243286970730, −1.91087614142896546948906405731, 0.25617412810522712179320350046, 1.74974948947432174235482274166, 3.43663021720279139065933442602, 4.03186768802987386861569880256, 5.05548514520747388913480249756, 5.64245245597497927151989567468, 6.91019048746134124423579093941, 7.45880652770112145326900834074, 8.458402058718272398987292675298, 8.949724444045713862190544591354

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