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Properties

Label 1-1375-1375.854-r1-0-0
Degree $1$
Conductor $1375$
Sign $0.670 - 0.742i$
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  + (0.728 + 0.684i)2-s + (−0.968 − 0.248i)3-s + (0.0627 + 0.998i)4-s + (−0.535 − 0.844i)6-s + (−0.809 − 0.587i)7-s + (−0.637 + 0.770i)8-s + (0.876 + 0.481i)9-s + (0.187 − 0.982i)12-s + (0.876 + 0.481i)13-s + (−0.187 − 0.982i)14-s + (−0.992 + 0.125i)16-s + (0.968 − 0.248i)17-s + (0.309 + 0.951i)18-s + (−0.535 − 0.844i)19-s + (0.637 + 0.770i)21-s + ⋯
L(s)  = 1  + (0.728 + 0.684i)2-s + (−0.968 − 0.248i)3-s + (0.0627 + 0.998i)4-s + (−0.535 − 0.844i)6-s + (−0.809 − 0.587i)7-s + (−0.637 + 0.770i)8-s + (0.876 + 0.481i)9-s + (0.187 − 0.982i)12-s + (0.876 + 0.481i)13-s + (−0.187 − 0.982i)14-s + (−0.992 + 0.125i)16-s + (0.968 − 0.248i)17-s + (0.309 + 0.951i)18-s + (−0.535 − 0.844i)19-s + (0.637 + 0.770i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.670 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.670 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.670 - 0.742i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (854, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ 0.670 - 0.742i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.037322850 - 0.4608047532i\)
\(L(\frac12)\) \(\approx\) \(1.037322850 - 0.4608047532i\)
\(L(1)\) \(\approx\) \(0.9628906503 + 0.2868206908i\)
\(L(1)\) \(\approx\) \(0.9628906503 + 0.2868206908i\)

Euler product

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

−21.05921892053336531669196558583, −20.1919583654742330726956584660, −19.08722487045999880441505012806, −18.68327980041182270249886679074, −17.95188434151618916587378614664, −16.72898469595522458888670677011, −16.229535477775989850563238455796, −15.246861642307910667012838992940, −14.876114510738454047507255160878, −13.54048140684687375788100579393, −12.93123011841786109933102048023, −12.21535207145778579484796640071, −11.7155172200206232073143569709, −10.63222770159258309894331507794, −10.261593235427063891101474977, −9.40677913885488478236168198508, −8.39797974056077037173966384819, −6.88197364432898201165312824933, −6.22280680105306720420034756553, −5.56131845632331474268063868765, −4.87776943786361487636563574267, −3.65324819604662373322457201958, −3.24603815659561930918448971450, −1.8268510258843891983908401581, −0.87250869312318061296795856607, 0.23843715918297763608985969102, 1.448319027067657569868607071972, 2.960839558307544645070940633521, 3.78726525389014241624518585637, 4.727506426558189543368701609119, 5.436588635639025821051106666319, 6.50053520453928977930994596905, 6.73637547096242669975717591725, 7.6182700701971486125739666020, 8.67008278334971583427110452531, 9.6847310687332711893158607795, 10.76352917940404967427372309546, 11.36417208157539088646679640190, 12.36729152902015128978141320598, 12.86265456696376487213552631226, 13.63632815872711145592200483280, 14.26282944088965210783409630201, 15.59769967538598525910593426425, 15.894173561016530120676344216936, 16.8449543760723635057509351534, 17.12600543927811703844661121072, 18.115716451062303488987901933926, 18.90044580976350627685961087346, 19.76489072388367434365197989579, 20.96165037129844520715609910176

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