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Properties

Label 1-1375-1375.277-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.952 - 0.306i$
Analytic cond. $6.38547$
Root an. cond. $6.38547$
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  + (−0.368 + 0.929i)2-s + (−0.125 + 0.992i)3-s + (−0.728 − 0.684i)4-s + (−0.876 − 0.481i)6-s + (−0.951 − 0.309i)7-s + (0.904 − 0.425i)8-s + (−0.968 − 0.248i)9-s + (0.770 − 0.637i)12-s + (−0.248 + 0.968i)13-s + (0.637 − 0.770i)14-s + (0.0627 + 0.998i)16-s + (0.125 + 0.992i)17-s + (0.587 − 0.809i)18-s + (0.876 + 0.481i)19-s + (0.425 − 0.904i)21-s + ⋯
L(s)  = 1  + (−0.368 + 0.929i)2-s + (−0.125 + 0.992i)3-s + (−0.728 − 0.684i)4-s + (−0.876 − 0.481i)6-s + (−0.951 − 0.309i)7-s + (0.904 − 0.425i)8-s + (−0.968 − 0.248i)9-s + (0.770 − 0.637i)12-s + (−0.248 + 0.968i)13-s + (0.637 − 0.770i)14-s + (0.0627 + 0.998i)16-s + (0.125 + 0.992i)17-s + (0.587 − 0.809i)18-s + (0.876 + 0.481i)19-s + (0.425 − 0.904i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.952 - 0.306i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.952 - 0.306i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2156652697 - 0.03381024277i\)
\(L(\frac12)\) \(\approx\) \(0.2156652697 - 0.03381024277i\)
\(L(1)\) \(\approx\) \(0.4393616046 + 0.3621821060i\)
\(L(1)\) \(\approx\) \(0.4393616046 + 0.3621821060i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.368 + 0.929i)T \)
3 \( 1 + (-0.125 + 0.992i)T \)
7 \( 1 + (-0.951 - 0.309i)T \)
13 \( 1 + (-0.248 + 0.968i)T \)
17 \( 1 + (0.125 + 0.992i)T \)
19 \( 1 + (0.876 + 0.481i)T \)
23 \( 1 + (-0.998 - 0.0627i)T \)
29 \( 1 + (-0.187 - 0.982i)T \)
31 \( 1 + (-0.425 - 0.904i)T \)
37 \( 1 + (-0.248 + 0.968i)T \)
41 \( 1 + (-0.968 - 0.248i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + (0.481 + 0.876i)T \)
53 \( 1 + (-0.481 - 0.876i)T \)
59 \( 1 + (-0.535 + 0.844i)T \)
61 \( 1 + (-0.535 - 0.844i)T \)
67 \( 1 + (-0.125 - 0.992i)T \)
71 \( 1 + (-0.992 - 0.125i)T \)
73 \( 1 + (0.368 - 0.929i)T \)
79 \( 1 + (-0.425 + 0.904i)T \)
83 \( 1 + (0.904 - 0.425i)T \)
89 \( 1 + (0.637 - 0.770i)T \)
97 \( 1 + (-0.684 + 0.728i)T \)
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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

−20.40417169103344800231994494078, −20.05844584324699141343805611288, −19.46566902319043077087452634432, −18.44596828062925203774903662063, −18.22976937625107248827398160199, −17.42344318001218338388350001650, −16.49333745832354655925352530144, −15.80214493527364171721334914040, −14.46112851833189051870162568997, −13.62031420515845925066158550754, −13.1038030608788811504962178541, −12.20415716820112119028301357965, −11.95226171188815832397556473931, −10.863185084174621614866354969512, −10.06962841294744056235138504677, −9.21396729420517158187976383656, −8.5072211240953253498296290385, −7.49187839225109440203563012102, −6.96588480489453272209742100776, −5.68597908573124491912526350407, −5.02825257789077685495534436283, −3.4334765546274804775934443060, −2.96464917173423829444888887845, −2.03656491719757247360409969954, −0.91845327036535822675989547921, 0.12031918246387755006647644436, 1.76566944392709497638350235259, 3.350824352389710590628239339540, 4.06215247812152684123643706796, 4.84815005752974013000604269787, 5.99844870949887017886527246402, 6.3026781397928127166633252091, 7.47661981566886282090996153405, 8.309819122609957239665555839067, 9.25161426012822167894687111299, 9.84405188827463887967121616812, 10.29109885464437921849155396424, 11.39244575857351521580927391962, 12.34220276223668042604826869020, 13.54384596471347879999896183855, 14.04965427046256482618570822387, 15.00511258373807724084872995563, 15.539998303131449914253395685, 16.463040390104268691211490935990, 16.68989040478630188837648810590, 17.452890741657077737597944642462, 18.51281463738295397136609715293, 19.2227834907090123860106804912, 19.95297255293653186285444403387, 20.759203065860057929732841715439

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