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Properties

Label 1-1375-1375.97-r1-0-0
Degree $1$
Conductor $1375$
Sign $0.975 + 0.219i$
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  + (0.982 − 0.187i)2-s + (−0.770 + 0.637i)3-s + (0.929 − 0.368i)4-s + (−0.637 + 0.770i)6-s + (−0.587 − 0.809i)7-s + (0.844 − 0.535i)8-s + (0.187 − 0.982i)9-s + (−0.481 + 0.876i)12-s + (0.982 + 0.187i)13-s + (−0.728 − 0.684i)14-s + (0.728 − 0.684i)16-s + (−0.844 + 0.535i)17-s − i·18-s + (−0.968 + 0.248i)19-s + (0.968 + 0.248i)21-s + ⋯
L(s)  = 1  + (0.982 − 0.187i)2-s + (−0.770 + 0.637i)3-s + (0.929 − 0.368i)4-s + (−0.637 + 0.770i)6-s + (−0.587 − 0.809i)7-s + (0.844 − 0.535i)8-s + (0.187 − 0.982i)9-s + (−0.481 + 0.876i)12-s + (0.982 + 0.187i)13-s + (−0.728 − 0.684i)14-s + (0.728 − 0.684i)16-s + (−0.844 + 0.535i)17-s − i·18-s + (−0.968 + 0.248i)19-s + (0.968 + 0.248i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.975 + 0.219i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ 0.975 + 0.219i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.842080427 + 0.3163252845i\)
\(L(\frac12)\) \(\approx\) \(2.842080427 + 0.3163252845i\)
\(L(1)\) \(\approx\) \(1.480568386 + 0.005785438335i\)
\(L(1)\) \(\approx\) \(1.480568386 + 0.005785438335i\)

Euler product

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

−20.83584884488808643908379102806, −19.86887107560578717352916554162, −19.13448723333596029384601378610, −18.3160892148882312045416130233, −17.566399811216526965055057045101, −16.69072007416620311460618222723, −15.825978654666913080089930116714, −15.58343899156568383564703807122, −14.382070078032316850177399275336, −13.51511249765141613710048112994, −12.91636610932765863806677320810, −12.38800857064769714006362059148, −11.433394208470720372130077279107, −11.053428093634561484392627933641, −9.9475851230159136509593959207, −8.66346006891226482582133430179, −7.87729768793896896893971310361, −6.83428299829741475787017445146, −6.164085947622972804734011378553, −5.78257559444634937175829506520, −4.69635323617383253249463843022, −3.88362129393686847287169090397, −2.56722730124618215303650565841, −2.0470617859449467813346392042, −0.58238238032108807645811064870, 0.72235533472555361345429404982, 1.81929468697435561656329807687, 3.20600923411647891482740449964, 4.002806067514665964563976223953, 4.40808721336761552478764990480, 5.53800682144091139023855790777, 6.42653307075162285361561316450, 6.66206940359840028711067297882, 8.00864537551720127839728901559, 9.25471424021592769470049060466, 10.152851280578552408422789400847, 10.904806187531007714629386592772, 11.16958037161209786746712690051, 12.44390956703673217808683287074, 12.83147885898451908355761079985, 13.782231753605657930797518943725, 14.54292509900106014826826296618, 15.42297019284319037822417153390, 16.18257980183992292158353092625, 16.48916111669467896883261356423, 17.5117777090736552688643273819, 18.3416541746499841855370314146, 19.58857870674927205762602689188, 19.99805009848394646285815141238, 20.94495714438440024898822397114

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