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Properties

Label 1-1375-1375.1139-r1-0-0
Degree $1$
Conductor $1375$
Sign $0.964 - 0.263i$
Analytic cond. $147.764$
Root an. cond. $147.764$
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  + (0.0627 − 0.998i)2-s + (0.187 + 0.982i)3-s + (−0.992 − 0.125i)4-s + (0.992 − 0.125i)6-s + (−0.809 + 0.587i)7-s + (−0.187 + 0.982i)8-s + (−0.929 + 0.368i)9-s + (−0.0627 − 0.998i)12-s + (−0.929 + 0.368i)13-s + (0.535 + 0.844i)14-s + (0.968 + 0.248i)16-s + (0.728 − 0.684i)17-s + (0.309 + 0.951i)18-s + (0.425 + 0.904i)19-s + (−0.728 − 0.684i)21-s + ⋯
L(s)  = 1  + (0.0627 − 0.998i)2-s + (0.187 + 0.982i)3-s + (−0.992 − 0.125i)4-s + (0.992 − 0.125i)6-s + (−0.809 + 0.587i)7-s + (−0.187 + 0.982i)8-s + (−0.929 + 0.368i)9-s + (−0.0627 − 0.998i)12-s + (−0.929 + 0.368i)13-s + (0.535 + 0.844i)14-s + (0.968 + 0.248i)16-s + (0.728 − 0.684i)17-s + (0.309 + 0.951i)18-s + (0.425 + 0.904i)19-s + (−0.728 − 0.684i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.964 - 0.263i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ 0.964 - 0.263i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.012726726 - 0.1360290136i\)
\(L(\frac12)\) \(\approx\) \(1.012726726 - 0.1360290136i\)
\(L(1)\) \(\approx\) \(0.7933276295 - 0.08408831736i\)
\(L(1)\) \(\approx\) \(0.7933276295 - 0.08408831736i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.0627 - 0.998i)T \)
3 \( 1 + (0.187 + 0.982i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.929 + 0.368i)T \)
17 \( 1 + (0.728 - 0.684i)T \)
19 \( 1 + (0.425 + 0.904i)T \)
23 \( 1 + (-0.535 - 0.844i)T \)
29 \( 1 + (-0.728 - 0.684i)T \)
31 \( 1 + (-0.425 - 0.904i)T \)
37 \( 1 + (-0.0627 - 0.998i)T \)
41 \( 1 + (-0.968 - 0.248i)T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (-0.728 - 0.684i)T \)
53 \( 1 + (0.992 + 0.125i)T \)
59 \( 1 + (0.968 + 0.248i)T \)
61 \( 1 + (-0.0627 + 0.998i)T \)
67 \( 1 + (0.425 + 0.904i)T \)
71 \( 1 + (-0.992 - 0.125i)T \)
73 \( 1 + (-0.929 - 0.368i)T \)
79 \( 1 + (-0.876 + 0.481i)T \)
83 \( 1 + (0.876 + 0.481i)T \)
89 \( 1 + (0.0627 - 0.998i)T \)
97 \( 1 + (0.187 + 0.982i)T \)
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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

−20.43661179741233302337853451853, −19.66517195557020051958531942541, −19.15333963068962952934732923494, −18.30504515961614679055823971450, −17.49580695145478440038911001767, −17.022604804612342255531044112986, −16.2210894048841652126115356471, −15.288740963748490751523390971156, −14.5367424976428549759797975740, −13.82021149751091437410309419838, −13.12153871880117850216444402577, −12.587698381965040630930641201490, −11.73697205800489752965435748201, −10.36388077486995254917032974603, −9.611401278930816915720478309170, −8.76932645293169285319220032946, −7.83976892894617312924369723787, −7.23168237799970153411489420932, −6.6822670828401027375076563693, −5.74057117282661873919889153539, −5.00838713251565046922586452381, −3.640053524452690900929047012438, −3.047520807833377687886653738, −1.539905630801153775272484188233, −0.45224237212903937705988386853, 0.40073691742840603185496069799, 2.07623494769579787817805981434, 2.7492366041635038862801244471, 3.61894384106304422423207306365, 4.32231676718163173074516574414, 5.367978012107174582736988512312, 5.88740965748761258443994209204, 7.434285708495241005695064680888, 8.4674727871565169111644283202, 9.25569148061386930657679412913, 9.91027036017482532704842512142, 10.231055099741342563806701598761, 11.52420376958411761316526145199, 11.92780596421047570928656182358, 12.80808970367587898903842052215, 13.72896091141127225456161223465, 14.57898558243003312316902126097, 15.014530265546697387723072288, 16.36122085487606051508673122359, 16.5488604594268476641578751948, 17.72643167969396433960577446445, 18.674990239526614728129290255715, 19.18853836477421401821495228682, 20.02505707292498261123893759122, 20.64611517791014499495714185187

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