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Properties

Label 1-1580-1580.1059-r1-0-0
Degree $1$
Conductor $1580$
Sign $-0.907 - 0.419i$
Analytic cond. $169.794$
Root an. cond. $169.794$
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  + (−0.0402 + 0.999i)3-s + (−0.845 − 0.534i)7-s + (−0.996 − 0.0804i)9-s + (0.919 − 0.391i)11-s + (0.200 + 0.979i)13-s + (0.748 − 0.663i)17-s + (−0.278 − 0.960i)19-s + (0.568 − 0.822i)21-s + (−0.5 − 0.866i)23-s + (0.120 − 0.992i)27-s + (0.428 − 0.903i)29-s + (0.632 − 0.774i)31-s + (0.354 + 0.935i)33-s + (−0.692 + 0.721i)37-s + (−0.987 + 0.160i)39-s + ⋯
L(s)  = 1  + (−0.0402 + 0.999i)3-s + (−0.845 − 0.534i)7-s + (−0.996 − 0.0804i)9-s + (0.919 − 0.391i)11-s + (0.200 + 0.979i)13-s + (0.748 − 0.663i)17-s + (−0.278 − 0.960i)19-s + (0.568 − 0.822i)21-s + (−0.5 − 0.866i)23-s + (0.120 − 0.992i)27-s + (0.428 − 0.903i)29-s + (0.632 − 0.774i)31-s + (0.354 + 0.935i)33-s + (−0.692 + 0.721i)37-s + (−0.987 + 0.160i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1580 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1580 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1580\)    =    \(2^{2} \cdot 5 \cdot 79\)
Sign: $-0.907 - 0.419i$
Analytic conductor: \(169.794\)
Root analytic conductor: \(169.794\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1580} (1059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1580,\ (1:\ ),\ -0.907 - 0.419i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006556510082 + 0.02982591837i\)
\(L(\frac12)\) \(\approx\) \(0.006556510082 + 0.02982591837i\)
\(L(1)\) \(\approx\) \(0.8495176581 + 0.1585081189i\)
\(L(1)\) \(\approx\) \(0.8495176581 + 0.1585081189i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
79 \( 1 \)
good3 \( 1 + (-0.0402 + 0.999i)T \)
7 \( 1 + (-0.845 - 0.534i)T \)
11 \( 1 + (0.919 - 0.391i)T \)
13 \( 1 + (0.200 + 0.979i)T \)
17 \( 1 + (0.748 - 0.663i)T \)
19 \( 1 + (-0.278 - 0.960i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.428 - 0.903i)T \)
31 \( 1 + (0.632 - 0.774i)T \)
37 \( 1 + (-0.692 + 0.721i)T \)
41 \( 1 + (0.120 + 0.992i)T \)
43 \( 1 + (-0.919 - 0.391i)T \)
47 \( 1 + (0.692 + 0.721i)T \)
53 \( 1 + (0.0402 + 0.999i)T \)
59 \( 1 + (-0.948 + 0.316i)T \)
61 \( 1 + (-0.970 - 0.239i)T \)
67 \( 1 + (-0.354 + 0.935i)T \)
71 \( 1 + (-0.885 - 0.464i)T \)
73 \( 1 + (0.200 - 0.979i)T \)
83 \( 1 + (0.948 + 0.316i)T \)
89 \( 1 + (0.885 - 0.464i)T \)
97 \( 1 + (0.970 + 0.239i)T \)
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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

−20.39895193339032674687356758693, −19.64938799191876898040397568688, −19.26231453674623753990007657961, −18.4253445254477581375531352959, −17.7341780656930437055307278265, −17.04857224560540830942890927550, −16.25120521467848546308146118794, −15.331452216457945030438766950228, −14.543063480347588299269741587675, −13.82332071673908363918182110534, −12.93198438355604792442684566036, −12.19927067842421510107366157700, −12.051129975528845091200426848068, −10.69361168124259997023264785726, −9.981040911167890720526264245722, −8.96703401023687524147209338283, −8.295986584557549976850444636, −7.44051820300295635482322639379, −6.58916067507835353040166335902, −5.93816479702719179676751296925, −5.27048247531633122930911369829, −3.68554374499944603215445751990, −3.16908521419973101098465889920, −1.95210512476962384280938987054, −1.22241692496119250227909026782, 0.00645619630305767258173933239, 1.01830844619132059549494274244, 2.57029390603899648051260277742, 3.353411454540019198752378408449, 4.23342982852030979003643901812, 4.731885375695232431251525407761, 6.165380242110193947121079392, 6.429956996437882743403581245324, 7.61999900596898803464954275070, 8.74692801711300666353290187064, 9.32014660936445858067168221036, 9.98516518182094660285769608897, 10.74125373034601180435527002464, 11.65816981631062262197463159238, 12.1356524510057229332515112383, 13.59926921318212094501376377757, 13.8513397399973862550475559641, 14.808144121169671524971353233096, 15.60445181793493151249988292714, 16.4034816405126611380984650844, 16.77178342182233233641585930153, 17.459754570179004400112208176259, 18.733878009501875385139104761943, 19.30747344153446569715268391850, 20.09779417311391109400920692294

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