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Properties

Label 1-1375-1375.1037-r1-0-0
Degree $1$
Conductor $1375$
Sign $0.262 - 0.965i$
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  + (0.844 + 0.535i)2-s + (−0.904 + 0.425i)3-s + (0.425 + 0.904i)4-s + (−0.992 − 0.125i)6-s + i·7-s + (−0.125 + 0.992i)8-s + (0.637 − 0.770i)9-s + (−0.770 − 0.637i)12-s + (0.770 + 0.637i)13-s + (−0.535 + 0.844i)14-s + (−0.637 + 0.770i)16-s + (−0.684 + 0.728i)17-s + (0.951 − 0.309i)18-s + (−0.728 − 0.684i)19-s + (−0.425 − 0.904i)21-s + ⋯
L(s)  = 1  + (0.844 + 0.535i)2-s + (−0.904 + 0.425i)3-s + (0.425 + 0.904i)4-s + (−0.992 − 0.125i)6-s + i·7-s + (−0.125 + 0.992i)8-s + (0.637 − 0.770i)9-s + (−0.770 − 0.637i)12-s + (0.770 + 0.637i)13-s + (−0.535 + 0.844i)14-s + (−0.637 + 0.770i)16-s + (−0.684 + 0.728i)17-s + (0.951 − 0.309i)18-s + (−0.728 − 0.684i)19-s + (−0.425 − 0.904i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.262 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.262 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.262 - 0.965i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1037, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ 0.262 - 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3955990043 + 0.3024700129i\)
\(L(\frac12)\) \(\approx\) \(-0.3955990043 + 0.3024700129i\)
\(L(1)\) \(\approx\) \(0.8346208138 + 0.7313819586i\)
\(L(1)\) \(\approx\) \(0.8346208138 + 0.7313819586i\)

Euler product

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

−20.12526767659565856715006888862, −19.45577664100226363043495785940, −18.517327192774140711376974212520, −17.955657277819742862363504065790, −16.92337505299270269949827737658, −16.28733580317100335621970984064, −15.4866836855000401765963002506, −14.51075381077538390322204513866, −13.66373882481055680042816727036, −13.00407457239976397107934853430, −12.58686735376312583276849208115, −11.402324758766946556383316219997, −10.96993826385494921152502519593, −10.40929585698138102510760952926, −9.43535278525105085535862075823, −8.06474625105380387161258689497, −7.02548547863404675889260322274, −6.541722400654329607232641479396, −5.57310868608767630303227880939, −4.83819021182831059242185844316, −4.010781124632360418824616571827, −3.09173471120860387812362969261, −1.797558682337832874973487228328, −1.02796134533633564992602852018, −0.08274643004988773767516249677, 1.66999605850109279719985818129, 2.7469420249284377204212109390, 3.89773556353511550117756206473, 4.55233975798082222013839823744, 5.37271750231304228507747520995, 6.30757169018793381373938294516, 6.482993654314155098318045540494, 7.78193769365157359574480990430, 8.81700125346963707381406712828, 9.40834011037307691535052375687, 10.92474464462795451086296675834, 11.252964630374250220438369037573, 12.03995065101738036925797052315, 12.960762209420436001024629615885, 13.339410518471863323593103610726, 14.81440452694540831535810560874, 15.143660750037603827592448292052, 15.7911674103085528801167521954, 16.71572133120786926064815509146, 17.11182606004758449352214846957, 18.12372185481571069592422281134, 18.72569949983606847296602856154, 19.93303154427956553718952815517, 20.96160003408694971261434369773, 21.59689984226245246086204246701

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