Properties

Label 1-1375-1375.123-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.950 + 0.312i$
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  + (−0.770 − 0.637i)2-s + (−0.684 + 0.728i)3-s + (0.187 + 0.982i)4-s + (0.992 − 0.125i)6-s + (0.951 + 0.309i)7-s + (0.481 − 0.876i)8-s + (−0.0627 − 0.998i)9-s + (−0.844 − 0.535i)12-s + (0.998 − 0.0627i)13-s + (−0.535 − 0.844i)14-s + (−0.929 + 0.368i)16-s + (0.684 + 0.728i)17-s + (−0.587 + 0.809i)18-s + (−0.992 + 0.125i)19-s + (−0.876 + 0.481i)21-s + ⋯
L(s)  = 1  + (−0.770 − 0.637i)2-s + (−0.684 + 0.728i)3-s + (0.187 + 0.982i)4-s + (0.992 − 0.125i)6-s + (0.951 + 0.309i)7-s + (0.481 − 0.876i)8-s + (−0.0627 − 0.998i)9-s + (−0.844 − 0.535i)12-s + (0.998 − 0.0627i)13-s + (−0.535 − 0.844i)14-s + (−0.929 + 0.368i)16-s + (0.684 + 0.728i)17-s + (−0.587 + 0.809i)18-s + (−0.992 + 0.125i)19-s + (−0.876 + 0.481i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.950 + 0.312i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.950 + 0.312i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9808267099 + 0.1569301521i\)
\(L(\frac12)\) \(\approx\) \(0.9808267099 + 0.1569301521i\)
\(L(1)\) \(\approx\) \(0.7387543282 + 0.02339967184i\)
\(L(1)\) \(\approx\) \(0.7387543282 + 0.02339967184i\)

Euler product

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

−20.75815792527463120386312572454, −19.756679225843605406311695188029, −19.03099524253213582768768936441, −18.290430022161798501586878942901, −17.89132558260712842198370135045, −16.9120060894359832086906577463, −16.71273637636688498091032419134, −15.56264491748015381307896937212, −14.89327082749650087102392347090, −13.77219256242556168859339547314, −13.48627713502472939137573311538, −12.136839571597242516564852614819, −11.23970341761791971434063989574, −10.964198961943874541334198852307, −9.92243744027502350165988954641, −8.93538006231850187480752906926, −7.85022994598730438239022875282, −7.73810580503132067325736852577, −6.57487849910919694605192291099, −5.96118620366939409312924639722, −5.11954630987155670969775225829, −4.26059268943571145366274974669, −2.50717789033644057495210807386, −1.48274445597051438993048853113, −0.80456088545495380938016547942, 0.888037265140138100655565543242, 1.81123538586100528861378913806, 3.04130666011699929489129105670, 3.99473458295359882528568648186, 4.68222319457779785871122965195, 5.81659839560778852099546108868, 6.610017440187710786353167597219, 7.844402859502954249324609832355, 8.60023440948550151958864917730, 9.16178579495302354265662605483, 10.318175532126418291398806846165, 10.79049088257549809408591730880, 11.30258457354085346679582429739, 12.30247735367707272063405423075, 12.73238866031874747237315270207, 14.09934273080457455281974394868, 14.98379643905984251537762687391, 15.73249649544275209644442459171, 16.6075744075124245399774557942, 17.12706337199812483742089857832, 17.89840543486439098255384332156, 18.48278579847333952255727082515, 19.25271149613066010535841300441, 20.3683993220384691959262618010, 21.003426296923129731923401596355

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