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Properties

Label 1-1375-1375.1004-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.710 + 0.703i$
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  + (0.425 − 0.904i)2-s + (−0.968 − 0.248i)3-s + (−0.637 − 0.770i)4-s + (−0.637 + 0.770i)6-s + (−0.309 + 0.951i)7-s + (−0.968 + 0.248i)8-s + (0.876 + 0.481i)9-s + (0.425 + 0.904i)12-s + (−0.876 − 0.481i)13-s + (0.728 + 0.684i)14-s + (−0.187 + 0.982i)16-s + (−0.535 − 0.844i)17-s + (0.809 − 0.587i)18-s + (0.0627 − 0.998i)19-s + (0.535 − 0.844i)21-s + ⋯
L(s)  = 1  + (0.425 − 0.904i)2-s + (−0.968 − 0.248i)3-s + (−0.637 − 0.770i)4-s + (−0.637 + 0.770i)6-s + (−0.309 + 0.951i)7-s + (−0.968 + 0.248i)8-s + (0.876 + 0.481i)9-s + (0.425 + 0.904i)12-s + (−0.876 − 0.481i)13-s + (0.728 + 0.684i)14-s + (−0.187 + 0.982i)16-s + (−0.535 − 0.844i)17-s + (0.809 − 0.587i)18-s + (0.0627 − 0.998i)19-s + (0.535 − 0.844i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.710 + 0.703i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1004, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.710 + 0.703i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2723015557 + 0.1120063078i\)
\(L(\frac12)\) \(\approx\) \(0.2723015557 + 0.1120063078i\)
\(L(1)\) \(\approx\) \(0.5921528893 - 0.3396296458i\)
\(L(1)\) \(\approx\) \(0.5921528893 - 0.3396296458i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.425 - 0.904i)T \)
3 \( 1 + (-0.968 - 0.248i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (-0.876 - 0.481i)T \)
17 \( 1 + (-0.535 - 0.844i)T \)
19 \( 1 + (0.0627 - 0.998i)T \)
23 \( 1 + (-0.728 - 0.684i)T \)
29 \( 1 + (0.535 - 0.844i)T \)
31 \( 1 + (0.0627 - 0.998i)T \)
37 \( 1 + (0.425 + 0.904i)T \)
41 \( 1 + (-0.187 + 0.982i)T \)
43 \( 1 + (0.809 - 0.587i)T \)
47 \( 1 + (-0.535 + 0.844i)T \)
53 \( 1 + (0.637 + 0.770i)T \)
59 \( 1 + (-0.187 + 0.982i)T \)
61 \( 1 + (-0.425 + 0.904i)T \)
67 \( 1 + (-0.0627 + 0.998i)T \)
71 \( 1 + (-0.637 - 0.770i)T \)
73 \( 1 + (-0.876 + 0.481i)T \)
79 \( 1 + (-0.929 + 0.368i)T \)
83 \( 1 + (0.929 + 0.368i)T \)
89 \( 1 + (-0.425 + 0.904i)T \)
97 \( 1 + (-0.968 - 0.248i)T \)
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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

−21.14239098444603561689596823672, −20.00916450790168032512841674808, −19.15338249912311483458470430663, −18.0468269616740220712590533461, −17.49945732623755496324075888317, −16.86065525879691696867689721105, −16.23526357878536072324149901901, −15.68186762699936535008173054176, −14.59721556681037431436902080833, −14.071695082192110122737862605381, −13.00791390420371978189553335552, −12.44963947343988901129395275893, −11.681950800410798794931496657519, −10.58597351117537968143835042494, −9.94944715410520521710428583251, −9.048894317751249731539617471106, −7.88299497087153372807010244670, −7.12313106659674853638068283544, −6.50152991796484785682928676282, −5.704706698631777293348676870613, −4.819445515788470838909096125458, −4.09044327847973335127988936528, −3.418109811955352726722871800572, −1.68051102854734904754057237753, −0.13397555832538498795942945832, 0.9701918666264894733485724227, 2.406580146271488410117683930600, 2.71224305424091201247260700454, 4.34057529757634053292572908537, 4.83949362827634173257161197608, 5.78145208024529282663556786444, 6.34598812250806735295926464426, 7.45370489575274467361166488299, 8.64838345261943596431713044189, 9.6206352814457750448400886453, 10.13474237965999180817256901969, 11.1746724812258593560026904898, 11.77933854365117516543675343200, 12.29052178718187637688575479675, 13.103095373624846663194075477349, 13.688524868310611576623074609762, 14.936199320701781372025637507306, 15.47631456191909806900885741167, 16.37555266936847090671032703700, 17.48332168324319926899806419824, 18.01455483900924327539301592947, 18.69637290861130953983607253726, 19.40382984850067869851329775447, 20.1572624873327373501205017272, 21.09742486366806517817540978593

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