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Properties

Label 1-1375-1375.31-r0-0-0
Degree $1$
Conductor $1375$
Sign $-0.902 + 0.429i$
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  + (0.876 + 0.481i)2-s + (0.535 + 0.844i)3-s + (0.535 + 0.844i)4-s + (0.0627 + 0.998i)6-s + 7-s + (0.0627 + 0.998i)8-s + (−0.425 + 0.904i)9-s + (−0.425 + 0.904i)12-s + (−0.425 + 0.904i)13-s + (0.876 + 0.481i)14-s + (−0.425 + 0.904i)16-s + (−0.929 + 0.368i)17-s + (−0.809 + 0.587i)18-s + (−0.929 + 0.368i)19-s + (0.535 + 0.844i)21-s + ⋯
L(s)  = 1  + (0.876 + 0.481i)2-s + (0.535 + 0.844i)3-s + (0.535 + 0.844i)4-s + (0.0627 + 0.998i)6-s + 7-s + (0.0627 + 0.998i)8-s + (−0.425 + 0.904i)9-s + (−0.425 + 0.904i)12-s + (−0.425 + 0.904i)13-s + (0.876 + 0.481i)14-s + (−0.425 + 0.904i)16-s + (−0.929 + 0.368i)17-s + (−0.809 + 0.587i)18-s + (−0.929 + 0.368i)19-s + (0.535 + 0.844i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.902 + 0.429i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ -0.902 + 0.429i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7121894506 + 3.152770998i\)
\(L(\frac12)\) \(\approx\) \(0.7121894506 + 3.152770998i\)
\(L(1)\) \(\approx\) \(1.487378489 + 1.471292066i\)
\(L(1)\) \(\approx\) \(1.487378489 + 1.471292066i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.876 + 0.481i)T \)
3 \( 1 + (0.535 + 0.844i)T \)
7 \( 1 + T \)
13 \( 1 + (-0.425 + 0.904i)T \)
17 \( 1 + (-0.929 + 0.368i)T \)
19 \( 1 + (-0.929 + 0.368i)T \)
23 \( 1 + (0.728 - 0.684i)T \)
29 \( 1 + (0.968 - 0.248i)T \)
31 \( 1 + (-0.637 - 0.770i)T \)
37 \( 1 + (0.728 + 0.684i)T \)
41 \( 1 + (0.876 - 0.481i)T \)
43 \( 1 + (-0.809 - 0.587i)T \)
47 \( 1 + (-0.637 + 0.770i)T \)
53 \( 1 + (0.0627 - 0.998i)T \)
59 \( 1 + (-0.992 - 0.125i)T \)
61 \( 1 + (-0.187 + 0.982i)T \)
67 \( 1 + (0.0627 + 0.998i)T \)
71 \( 1 + (0.968 - 0.248i)T \)
73 \( 1 + (0.728 - 0.684i)T \)
79 \( 1 + (0.968 - 0.248i)T \)
83 \( 1 + (0.968 + 0.248i)T \)
89 \( 1 + (-0.425 - 0.904i)T \)
97 \( 1 + (-0.637 + 0.770i)T \)
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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

−20.382851208838431268435175881773, −19.8518045816872119003558423002, −19.37048460570998725675098004085, −18.150502831482653797042165091373, −17.89927510706212289605741351242, −16.80239530384005013393688939915, −15.3804577077935138792413192176, −15.09186483217645775142049734879, −14.22725000463555407090551719253, −13.638537034124777074594845495777, −12.78549807746898092393742000352, −12.34717955380800741932212805652, −11.236367583818905758774145160360, −10.89781886854694353899948860737, −9.6372047271122907970196103059, −8.74054137946591755330928585879, −7.831842486967984300730288336278, −7.03571424789703396545819712287, −6.24530654507390293156750960482, −5.192348343169143093525813480036, −4.523863217524471867896142697108, −3.35459476535667309173889345832, −2.54512771730157390737412497044, −1.80753799911353158938199539365, −0.80125368113945985447633578467, 1.95106237987811279340974728064, 2.51162891118393041166430020823, 3.74678079542835385626217258742, 4.53098259695638822757784233230, 4.83033940257002045782885703394, 6.01686254765246780682362223919, 6.90015821422522458524006888697, 7.947184174183696396471138844779, 8.5048683522481665793947651671, 9.29937721676150921357774485923, 10.57795551209487648399573740616, 11.139826568524655530143381760413, 11.932378605561721043251630512911, 12.96961348409940930677316758530, 13.74871776026499321141876153447, 14.53541417156573502880914347006, 14.89668416977024987281079597980, 15.592000857452825158743598159475, 16.61461481090349722122062283197, 16.981124729401979464794594879277, 17.91394075496654957042509315957, 19.11550521622504323361356305828, 19.916723948001344959559959696877, 20.776666355236455974348202036703, 21.19253856198962136479738737867

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