Properties

Label 1-1375-1375.1069-r1-0-0
Degree $1$
Conductor $1375$
Sign $0.993 + 0.115i$
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  + (−0.992 + 0.125i)2-s + (−0.968 + 0.248i)3-s + (0.968 − 0.248i)4-s + (0.929 − 0.368i)6-s + 7-s + (−0.929 + 0.368i)8-s + (0.876 − 0.481i)9-s + (−0.876 + 0.481i)12-s + (0.876 − 0.481i)13-s + (−0.992 + 0.125i)14-s + (0.876 − 0.481i)16-s + (−0.637 − 0.770i)17-s + (−0.809 + 0.587i)18-s + (0.637 + 0.770i)19-s + (−0.968 + 0.248i)21-s + ⋯
L(s)  = 1  + (−0.992 + 0.125i)2-s + (−0.968 + 0.248i)3-s + (0.968 − 0.248i)4-s + (0.929 − 0.368i)6-s + 7-s + (−0.929 + 0.368i)8-s + (0.876 − 0.481i)9-s + (−0.876 + 0.481i)12-s + (0.876 − 0.481i)13-s + (−0.992 + 0.125i)14-s + (0.876 − 0.481i)16-s + (−0.637 − 0.770i)17-s + (−0.809 + 0.587i)18-s + (0.637 + 0.770i)19-s + (−0.968 + 0.248i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.993 + 0.115i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1069, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ 0.993 + 0.115i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.262825391 + 0.07348093810i\)
\(L(\frac12)\) \(\approx\) \(1.262825391 + 0.07348093810i\)
\(L(1)\) \(\approx\) \(0.6808398626 + 0.04452164913i\)
\(L(1)\) \(\approx\) \(0.6808398626 + 0.04452164913i\)

Euler product

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

−20.65991282817103246113630235161, −19.63710294558511645382887276768, −19.02445961988687574952572194436, −18.12390947688926631842295475024, −17.60116309000413230295491754253, −17.27169812043735826083235436710, −16.101715821611293689646130813865, −15.75483142874087732608510169358, −14.753154754672027433976088590630, −13.58387261717090092647655502498, −12.787142826205613325691111980142, −11.68854492584754550548593918289, −11.34199413433438735783150249514, −10.777791963120300495822915436837, −9.82543821089630129634334332478, −8.90731570061688205396858507733, −8.075009594836396134935035304306, −7.3276019711088796953516740679, −6.49985742091233660152640566152, −5.748425760905143285605435579628, −4.73656858946846030756018907285, −3.70261602564866286329963368462, −2.22975080843178100076031971662, −1.46532837404533562198329980678, −0.67065048952916769831618037157, 0.671589496637821573631535459153, 1.30511016389266639630190683503, 2.472326626411725977009559480004, 3.7903240817123806989797935419, 4.933799966520684513683575679407, 5.63171849034251600723160154556, 6.51459691655404698002785545981, 7.29098352515076311924799539523, 8.209736704884752953524337731906, 8.90528515817984638760920086754, 9.976334518474403153688945711381, 10.552729095296236008725146220521, 11.37142963174018560291474684286, 11.72738300967947160226322226482, 12.73860102105942133182257675437, 13.88752241534920241459413676440, 14.95766515988298973711387877147, 15.50069150339021633799365662111, 16.43397645793672483014587701242, 16.80630713031077568002002129242, 17.87324798176573356447074878030, 18.18488699466275933056954286603, 18.69906660731995492771261593166, 20.042629421020453740254507225953, 20.67045100558761621629014090160

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