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Properties

Label 1-1375-1375.1096-r1-0-0
Degree $1$
Conductor $1375$
Sign $-0.106 + 0.994i$
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  + (0.425 − 0.904i)2-s + (−0.929 + 0.368i)3-s + (−0.637 − 0.770i)4-s + (−0.0627 + 0.998i)6-s + (0.809 + 0.587i)7-s + (−0.968 + 0.248i)8-s + (0.728 − 0.684i)9-s + (0.876 + 0.481i)12-s + (−0.728 + 0.684i)13-s + (0.876 − 0.481i)14-s + (−0.187 + 0.982i)16-s + (0.929 + 0.368i)17-s + (−0.309 − 0.951i)18-s + (−0.0627 + 0.998i)19-s + (−0.968 − 0.248i)21-s + ⋯
L(s)  = 1  + (0.425 − 0.904i)2-s + (−0.929 + 0.368i)3-s + (−0.637 − 0.770i)4-s + (−0.0627 + 0.998i)6-s + (0.809 + 0.587i)7-s + (−0.968 + 0.248i)8-s + (0.728 − 0.684i)9-s + (0.876 + 0.481i)12-s + (−0.728 + 0.684i)13-s + (0.876 − 0.481i)14-s + (−0.187 + 0.982i)16-s + (0.929 + 0.368i)17-s + (−0.309 − 0.951i)18-s + (−0.0627 + 0.998i)19-s + (−0.968 − 0.248i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.106 + 0.994i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1096, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ -0.106 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5790177136 + 0.6441489706i\)
\(L(\frac12)\) \(\approx\) \(0.5790177136 + 0.6441489706i\)
\(L(1)\) \(\approx\) \(0.8834028994 - 0.1799695299i\)
\(L(1)\) \(\approx\) \(0.8834028994 - 0.1799695299i\)

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 \( 1 + (-0.929 + 0.368i)T \)
7 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (-0.728 + 0.684i)T \)
17 \( 1 + (0.929 + 0.368i)T \)
19 \( 1 + (-0.0627 + 0.998i)T \)
23 \( 1 + (-0.187 - 0.982i)T \)
29 \( 1 + (-0.535 + 0.844i)T \)
31 \( 1 + (0.968 - 0.248i)T \)
37 \( 1 + (0.728 - 0.684i)T \)
41 \( 1 + (-0.728 + 0.684i)T \)
43 \( 1 + (0.809 - 0.587i)T \)
47 \( 1 + (0.0627 + 0.998i)T \)
53 \( 1 + (0.0627 + 0.998i)T \)
59 \( 1 + (-0.992 + 0.125i)T \)
61 \( 1 + (0.992 + 0.125i)T \)
67 \( 1 + (-0.929 - 0.368i)T \)
71 \( 1 + (-0.929 + 0.368i)T \)
73 \( 1 + (0.425 - 0.904i)T \)
79 \( 1 + (-0.968 - 0.248i)T \)
83 \( 1 + (-0.968 + 0.248i)T \)
89 \( 1 + (0.876 - 0.481i)T \)
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

−20.704307626225543817339355172685, −19.58884099034484683493163842255, −18.63689341397736666283136084627, −17.81636535599362989558229197855, −17.30309949652267608262838796446, −16.88187374517860525484776604497, −15.88774186073028277825534290722, −15.21546929685878050159245976940, −14.35448375205204625410510171456, −13.504882585759632925489929536252, −12.99855652046559231681998880022, −11.88422585149504795729217810200, −11.56136773042573917523482895899, −10.35549673191767277576456234239, −9.61568556296676672156890991494, −8.28027957981584273563145335593, −7.584018649711243382009181695770, −7.12371707275969335634644118389, −6.11246105973261969378663164011, −5.241363276576952013847042578673, −4.81739065887123053792004237693, −3.82365557607997499709634088053, −2.56078358835433414066018042623, −1.10682340939155995469488008733, −0.193788921769643484255432326968, 1.11645721219439397895219906388, 1.90905529805494285766892549007, 3.03405014511081084073757408218, 4.23564416171055997230313316113, 4.66972215224525941639560254007, 5.663069586649772252108472271095, 6.12521168076461319355765262357, 7.45862943794308905022107628960, 8.587457003670022814788835227180, 9.48368851197644984116917869091, 10.21863196363112694174724407727, 10.89677305079226715271050169753, 11.71742856432304546636042488967, 12.24087966994782342162219895484, 12.748775922419989356710047001, 14.1484922155321454389917312960, 14.605899130552609609874241139851, 15.337849036804363486181507991196, 16.45305976353584547010206902620, 17.108962945853641137964586380195, 18.023858564474686917258915941814, 18.62636468264355969246875840565, 19.20951903835898271478877280264, 20.46635493157638593295182181016, 20.95863696054710666423924350461

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