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Properties

Label 2-1375-275.241-c0-0-0
Degree $2$
Conductor $1375$
Sign $-0.728 - 0.684i$
Analytic cond. $0.686214$
Root an. cond. $0.828380$
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  + (0.5 + 1.53i)3-s + (0.309 + 0.951i)4-s + (−1.30 + 0.951i)9-s + (−0.809 − 0.587i)11-s + (−1.30 + 0.951i)12-s + (−0.809 + 0.587i)16-s + (0.5 + 0.363i)23-s + (−0.809 − 0.587i)27-s + (−0.5 + 1.53i)31-s + (0.5 − 1.53i)33-s + (−1.30 − 0.951i)36-s + (1.61 − 1.17i)37-s + (0.309 − 0.951i)44-s + (−0.618 − 1.90i)47-s + (−1.30 − 0.951i)48-s + 49-s + ⋯
L(s)  = 1  + (0.5 + 1.53i)3-s + (0.309 + 0.951i)4-s + (−1.30 + 0.951i)9-s + (−0.809 − 0.587i)11-s + (−1.30 + 0.951i)12-s + (−0.809 + 0.587i)16-s + (0.5 + 0.363i)23-s + (−0.809 − 0.587i)27-s + (−0.5 + 1.53i)31-s + (0.5 − 1.53i)33-s + (−1.30 − 0.951i)36-s + (1.61 − 1.17i)37-s + (0.309 − 0.951i)44-s + (−0.618 − 1.90i)47-s + (−1.30 − 0.951i)48-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.728 - 0.684i$
Analytic conductor: \(0.686214\)
Root analytic conductor: \(0.828380\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ -0.728 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.247885426\)
\(L(\frac12)\) \(\approx\) \(1.247885426\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-0.309 - 0.951i)T^{2} \)
3 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 - T^{2} \)
13 \( 1 + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-1.61 + 1.17i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.618 + 1.90i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.12788701024786254399477423793, −9.112485004485383389300672155543, −8.676596639512623058065413126560, −7.86991897333301906057352358157, −7.02461831912572039996735038258, −5.71168018573241265125975884398, −4.87854641976471179791419807756, −3.92218928637781385145526335428, −3.26483745746788834964761254346, −2.44959339200237007346547782085, 1.00564837595745383167893861760, 2.15162910204939215391922770890, 2.77028123674227955906554159367, 4.47868078135867161399493826623, 5.58791178950164515768441678157, 6.31175000788136898618653575095, 7.09566068646308460201644696513, 7.69423879389020271016716350362, 8.483603418659264196630774225719, 9.497486294506470096974076948268

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