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Properties

Label 1-1375-1375.552-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.00314 - 0.999i$
Analytic cond. $6.38547$
Root an. cond. $6.38547$
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  + (−0.998 − 0.0627i)2-s + (−0.481 − 0.876i)3-s + (0.992 + 0.125i)4-s + (0.425 + 0.904i)6-s + (−0.951 − 0.309i)7-s + (−0.982 − 0.187i)8-s + (−0.535 + 0.844i)9-s + (−0.368 − 0.929i)12-s + (0.844 + 0.535i)13-s + (0.929 + 0.368i)14-s + (0.968 + 0.248i)16-s + (0.481 − 0.876i)17-s + (0.587 − 0.809i)18-s + (−0.425 − 0.904i)19-s + (0.187 + 0.982i)21-s + ⋯
L(s)  = 1  + (−0.998 − 0.0627i)2-s + (−0.481 − 0.876i)3-s + (0.992 + 0.125i)4-s + (0.425 + 0.904i)6-s + (−0.951 − 0.309i)7-s + (−0.982 − 0.187i)8-s + (−0.535 + 0.844i)9-s + (−0.368 − 0.929i)12-s + (0.844 + 0.535i)13-s + (0.929 + 0.368i)14-s + (0.968 + 0.248i)16-s + (0.481 − 0.876i)17-s + (0.587 − 0.809i)18-s + (−0.425 − 0.904i)19-s + (0.187 + 0.982i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00314 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00314 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.00314 - 0.999i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (552, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.00314 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4420677657 - 0.4406787447i\)
\(L(\frac12)\) \(\approx\) \(0.4420677657 - 0.4406787447i\)
\(L(1)\) \(\approx\) \(0.5195332180 - 0.1967447978i\)
\(L(1)\) \(\approx\) \(0.5195332180 - 0.1967447978i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.998 - 0.0627i)T \)
3 \( 1 + (-0.481 - 0.876i)T \)
7 \( 1 + (-0.951 - 0.309i)T \)
13 \( 1 + (0.844 + 0.535i)T \)
17 \( 1 + (0.481 - 0.876i)T \)
19 \( 1 + (-0.425 - 0.904i)T \)
23 \( 1 + (-0.248 - 0.968i)T \)
29 \( 1 + (0.728 + 0.684i)T \)
31 \( 1 + (-0.187 + 0.982i)T \)
37 \( 1 + (0.844 + 0.535i)T \)
41 \( 1 + (-0.535 + 0.844i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + (-0.904 - 0.425i)T \)
53 \( 1 + (0.904 + 0.425i)T \)
59 \( 1 + (0.637 + 0.770i)T \)
61 \( 1 + (0.637 - 0.770i)T \)
67 \( 1 + (-0.481 + 0.876i)T \)
71 \( 1 + (0.876 - 0.481i)T \)
73 \( 1 + (0.998 + 0.0627i)T \)
79 \( 1 + (-0.187 - 0.982i)T \)
83 \( 1 + (-0.982 - 0.187i)T \)
89 \( 1 + (0.929 + 0.368i)T \)
97 \( 1 + (0.125 - 0.992i)T \)
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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

−21.09491614691178361553429412195, −20.17532683077345391250839870020, −19.49165197243771086558296414572, −18.6943998439236855099200731790, −17.970244485472537938584004817435, −17.09448338779875224747516783408, −16.582260879293521640663563464644, −15.8120342592357885920306659601, −15.33552442340157962784869843106, −14.567086770187182222708101231987, −13.21009243257684712200550147089, −12.31184397348043911487680052921, −11.58645529943655705706503081610, −10.740522493518589463394563326750, −10.01549200741259833435678659986, −9.61758219318954756667814931123, −8.59262584907224353493353385366, −7.99437926848040989293228751101, −6.68520743619258040974037181357, −5.97397455447800512035402524212, −5.51930037128386032705434900300, −3.85093008484592596359865784688, −3.37782909966027389139644645821, −2.137774036324868887858582425770, −0.79106260629589508373836002755, 0.532995544376548454004014593939, 1.4062452306351067122462876357, 2.55992088272417853430711599735, 3.29691449596177227535070522380, 4.78836921833526059375647058047, 5.999525261187067265488200594517, 6.75133628745666722485380105971, 7.00924442121918380989958904272, 8.2184863919771888715352983734, 8.80228984255140411138201808174, 9.822584883015969367463358060888, 10.56557246613425840688179381354, 11.36973681787757912553505541699, 12.03021186633012549136155356531, 12.87424022462832861432255096925, 13.55412685045226165188974662560, 14.54576786566731045368658227638, 15.74769910987598789585907304552, 16.502952934731709836468356468334, 16.70654666902442248386954690250, 17.92142761277372974550619152501, 18.309956610942824012348454803656, 19.00291652936587845503940907876, 19.77310163935177816744112013304, 20.21928460896691783887135137549

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