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Properties

Label 1-1375-1375.1083-r1-0-0
Degree $1$
Conductor $1375$
Sign $-0.659 + 0.751i$
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  + (0.481 − 0.876i)2-s + (0.248 + 0.968i)3-s + (−0.535 − 0.844i)4-s + (0.968 + 0.248i)6-s + (−0.587 + 0.809i)7-s + (−0.998 + 0.0627i)8-s + (−0.876 + 0.481i)9-s + (0.684 − 0.728i)12-s + (0.481 + 0.876i)13-s + (0.425 + 0.904i)14-s + (−0.425 + 0.904i)16-s + (0.998 − 0.0627i)17-s + i·18-s + (0.929 − 0.368i)19-s + (−0.929 − 0.368i)21-s + ⋯
L(s)  = 1  + (0.481 − 0.876i)2-s + (0.248 + 0.968i)3-s + (−0.535 − 0.844i)4-s + (0.968 + 0.248i)6-s + (−0.587 + 0.809i)7-s + (−0.998 + 0.0627i)8-s + (−0.876 + 0.481i)9-s + (0.684 − 0.728i)12-s + (0.481 + 0.876i)13-s + (0.425 + 0.904i)14-s + (−0.425 + 0.904i)16-s + (0.998 − 0.0627i)17-s + i·18-s + (0.929 − 0.368i)19-s + (−0.929 − 0.368i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.659 + 0.751i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1083, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ -0.659 + 0.751i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6066131333 + 1.340322110i\)
\(L(\frac12)\) \(\approx\) \(0.6066131333 + 1.340322110i\)
\(L(1)\) \(\approx\) \(1.179888510 + 0.09025260694i\)
\(L(1)\) \(\approx\) \(1.179888510 + 0.09025260694i\)

Euler product

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

−20.55483340223501973180981732921, −19.53286880527642531836619009417, −18.72715483071776276464737597427, −17.99905626282584921597907377429, −17.33135129634221833374187819654, −16.531782671057613427008128295793, −15.90249159911064016530315133313, −14.877619524722974288557174371526, −14.0931033258375586560279783458, −13.67133913520040204802462798451, −12.72962505241636456757320847861, −12.42642938531991608541910370586, −11.328372605079797685715464091740, −10.154291115706809154431501808959, −9.2504065807661822562816734644, −8.22712270564963296754809901524, −7.62121815941137732040676716935, −7.08861391058937241465159131515, −5.966475608109175997610260512506, −5.71213212452616545385390139303, −4.23855527882114345779284143564, −3.41660431065603738892974751676, −2.692498742788247617189576616978, −1.10096276573734626704609699880, −0.25481160917599986563096915783, 1.26560578481963936349809344249, 2.42629508585425978738801135300, 3.22252459664149217811187685054, 3.828115391130888703015867912456, 4.85151439333396876002390891807, 5.591119097140679287255262491006, 6.27997828791257818061061717777, 7.76648768403824851575607749251, 8.992897545693702312148559986004, 9.31772603306239553570685689846, 10.05086443358970551349677742122, 10.89688734292083040072199185990, 11.809694413788524867321006882405, 12.15138521188563056627929201356, 13.50411952457945949470339620749, 13.86515874323986048788009601364, 14.83051351504768788111138211370, 15.50817675839786546700912211629, 16.130214884805805594757578802946, 17.05331795075620421006820828601, 18.22821707372737839293113967267, 18.87581150581327284150586208725, 19.55469773320494049947305931593, 20.325765953733137242933986277530, 21.0373312486353950563860792028

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