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Properties

Label 1-1375-1375.948-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.00314 + 0.999i$
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  + (0.998 + 0.0627i)2-s + (0.481 + 0.876i)3-s + (0.992 + 0.125i)4-s + (0.425 + 0.904i)6-s + (0.951 + 0.309i)7-s + (0.982 + 0.187i)8-s + (−0.535 + 0.844i)9-s + (0.368 + 0.929i)12-s + (−0.844 − 0.535i)13-s + (0.929 + 0.368i)14-s + (0.968 + 0.248i)16-s + (−0.481 + 0.876i)17-s + (−0.587 + 0.809i)18-s + (−0.425 − 0.904i)19-s + (0.187 + 0.982i)21-s + ⋯
L(s)  = 1  + (0.998 + 0.0627i)2-s + (0.481 + 0.876i)3-s + (0.992 + 0.125i)4-s + (0.425 + 0.904i)6-s + (0.951 + 0.309i)7-s + (0.982 + 0.187i)8-s + (−0.535 + 0.844i)9-s + (0.368 + 0.929i)12-s + (−0.844 − 0.535i)13-s + (0.929 + 0.368i)14-s + (0.968 + 0.248i)16-s + (−0.481 + 0.876i)17-s + (−0.587 + 0.809i)18-s + (−0.425 − 0.904i)19-s + (0.187 + 0.982i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00314 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00314 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.00314 + 0.999i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (948, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.00314 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.709839644 + 2.718381069i\)
\(L(\frac12)\) \(\approx\) \(2.709839644 + 2.718381069i\)
\(L(1)\) \(\approx\) \(2.145676940 + 1.028766503i\)
\(L(1)\) \(\approx\) \(2.145676940 + 1.028766503i\)

Euler product

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

−20.554842039895145059670958214117, −20.24679901008845562148809946724, −19.09564968115758884763716828159, −18.724081629860837692910710950149, −17.469086332710615949691682757925, −17.00304303150195785549560623987, −15.88823708811992469579396550870, −14.93283030623108363907897065149, −14.35706223100414866926676730608, −13.89207425495092520927435501219, −13.087795710376167352488149052977, −12.17401955415237969787253838070, −11.7720686250894356738733658931, −10.87247066503961104572875760254, −9.88986402109727595035600195227, −8.690422614509062742321941913824, −7.86244093819174470746128316292, −7.15171170605963702466595890180, −6.504458482811414269062698474236, −5.46720416457166794935307149757, −4.556100599904121005475957593333, −3.81793164181273962246523123005, −2.49092333302710747390482595004, −2.13610944216982651213971852172, −0.96080398621985212171081505665, 1.672769288573847529351377847404, 2.51990779905453871970541459400, 3.34168094012053357627943862922, 4.3163524947245001140491495, 5.00236722561446939279150265066, 5.51608196759247000483890479951, 6.76282857117918307437370280881, 7.71310134616059341998128829896, 8.44702268931350395965099166320, 9.32777047581858120319224547550, 10.60755680206799905465148588428, 10.82950770253547350529666409931, 11.85052169391513362883119611306, 12.65780792104242278138289981356, 13.5642337403776019669423460142, 14.34044480705701622242151657200, 14.90648273872433648631289246034, 15.468003754552983369024944816219, 16.10652864618115837482496625392, 17.31638200539623684112406215680, 17.56141303986630466080067848830, 19.23094561739146195554640268685, 19.74222780679708414759757234139, 20.45064993333659564393195122562, 21.24554942422451374182445424767

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