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Properties

Label 1-775-775.29-r1-0-0
Degree $1$
Conductor $775$
Sign $-0.226 + 0.974i$
Analytic cond. $83.2853$
Root an. cond. $83.2853$
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  + (−0.309 + 0.951i)2-s + 3-s + (−0.809 − 0.587i)4-s + (−0.309 + 0.951i)6-s + (0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + 9-s − 11-s + (−0.809 − 0.587i)12-s + (0.309 + 0.951i)13-s + (0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s + (−0.809 + 0.587i)17-s + (−0.309 + 0.951i)18-s + 19-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + 3-s + (−0.809 − 0.587i)4-s + (−0.309 + 0.951i)6-s + (0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + 9-s − 11-s + (−0.809 − 0.587i)12-s + (0.309 + 0.951i)13-s + (0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s + (−0.809 + 0.587i)17-s + (−0.309 + 0.951i)18-s + 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.226 + 0.974i$
Analytic conductor: \(83.2853\)
Root analytic conductor: \(83.2853\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 775,\ (1:\ ),\ -0.226 + 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.593235033 + 2.006094694i\)
\(L(\frac12)\) \(\approx\) \(1.593235033 + 2.006094694i\)
\(L(1)\) \(\approx\) \(1.179229345 + 0.6152517273i\)
\(L(1)\) \(\approx\) \(1.179229345 + 0.6152517273i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + T \)
7 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 - T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 - T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + T \)
89 \( 1 - T \)
97 \( 1 + (0.809 + 0.587i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.67388840401767791776165675293, −20.78414987730873473503232646161, −20.4737366862910256747078048568, −19.71060071113166697857007905209, −18.544197947315416123441932358993, −18.29251696897994030394970318103, −17.53821818945113781838018265309, −16.0593380882002377212239121410, −15.41236677854229491676250362322, −14.36854162100042074470242742457, −13.647514263548959777122832657136, −12.86417431994553006057615435149, −12.1161277559507462179723945349, −10.998236067723153043920747316290, −10.35558415456104441029089758279, −9.38520158235766212792670642063, −8.50042883012687850904822166735, −8.09152774117406178376421137806, −7.09860533374877981292935459167, −5.27428343696278790493659744608, −4.666292483192592571275221890, −3.333094912250918773364894775219, −2.67645809444428604916522261517, −1.84105406700473825188031830248, −0.62236086401877225842469228119, 1.087390364037673540637379602268, 2.0367301841277087954703355311, 3.54307446534229398574874002238, 4.47535812070590561737888545603, 5.24164338268247608241717685818, 6.63531259965319228887017162444, 7.36744609799173412399317809371, 8.13592166503303794292437871265, 8.73103463125835182544611094209, 9.74014425192185704928055763576, 10.46130492511676396245599236540, 11.554703387329882645274655402111, 13.10661905822845568281470387560, 13.6407281722126412969328560495, 14.232423376008585707969285475161, 15.176934317557808035585989450103, 15.73780702510406919961614876348, 16.59046126032072386405815220590, 17.70380373585595517553911165420, 18.166630538684352610454236974136, 19.17014821771498283873148671027, 19.76513882221716783666382418108, 20.85974692262708091083118237315, 21.36789735320637207085298068637, 22.525250962404863239667675092960

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