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Properties

Label 1-1375-1375.1173-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.0443 + 0.999i$
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  + (0.368 − 0.929i)2-s + (−0.684 + 0.728i)3-s + (−0.728 − 0.684i)4-s + (0.425 + 0.904i)6-s + i·7-s + (−0.904 + 0.425i)8-s + (−0.0627 − 0.998i)9-s + (0.998 − 0.0627i)12-s + (0.998 − 0.0627i)13-s + (0.929 + 0.368i)14-s + (0.0627 + 0.998i)16-s + (−0.481 + 0.876i)17-s + (−0.951 − 0.309i)18-s + (0.876 + 0.481i)19-s + (−0.728 − 0.684i)21-s + ⋯
L(s)  = 1  + (0.368 − 0.929i)2-s + (−0.684 + 0.728i)3-s + (−0.728 − 0.684i)4-s + (0.425 + 0.904i)6-s + i·7-s + (−0.904 + 0.425i)8-s + (−0.0627 − 0.998i)9-s + (0.998 − 0.0627i)12-s + (0.998 − 0.0627i)13-s + (0.929 + 0.368i)14-s + (0.0627 + 0.998i)16-s + (−0.481 + 0.876i)17-s + (−0.951 − 0.309i)18-s + (0.876 + 0.481i)19-s + (−0.728 − 0.684i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0443 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0443 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.0443 + 0.999i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.0443 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5567946711 + 0.5326055750i\)
\(L(\frac12)\) \(\approx\) \(0.5567946711 + 0.5326055750i\)
\(L(1)\) \(\approx\) \(0.8389582186 - 0.04790553947i\)
\(L(1)\) \(\approx\) \(0.8389582186 - 0.04790553947i\)

Euler product

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

−20.61323728305126961065422015976, −19.984821275799091559995438984873, −18.804216326572057534292093182288, −18.038158913810779574821402149694, −17.74759446818268993251802899035, −16.68167383590331482590434982466, −16.26279385561302880031981104226, −15.59447602872325415540258581178, −14.19322353221760486748112720034, −13.89085091914861519257966011464, −13.12835067514481562889576267490, −12.46008893428464716174390870234, −11.441335546994686110872619286608, −10.82758886351043079709714026840, −9.6641706739891234963932036073, −8.6685413571627387017399268740, −7.76360033590928637253725944829, −7.117347479920485997948591001733, −6.54310589153171090228111557538, −5.65364154396090100546394417389, −4.85166229553497235885970860372, −4.00338574668355810655488968228, −2.95285242583865153263140894292, −1.4441112172978706050618862021, −0.31545849432059874379017339631, 1.26784234328598424439497956587, 2.2701896918029124373238073875, 3.49276798973324695786123718044, 3.97023784118559454156303245884, 5.04849444544057998956169492090, 5.882039688988502496810854347073, 6.15179928049439394721514774954, 7.904768423098003846294146253460, 9.045963228047613189730363880676, 9.35870509576513812132649359094, 10.46260564771383841684103052110, 10.96990277308409493081269242537, 11.811362990804540140166409018518, 12.307061961550894920184327920157, 13.17248395205832961850647263748, 14.1403971924470524683012301829, 14.96363660553144931272931601162, 15.67904923081806337716786480304, 16.255193558486407666278797937413, 17.685987117826283232222869842047, 17.87034742509546837619382071720, 18.83312680280082998659258037872, 19.56019704406589794224697517474, 20.6282834639560326327844105458, 21.03020485282323821970452447910

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