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Properties

Label 2-1375-1375.1044-c0-0-0
Degree $2$
Conductor $1375$
Sign $-0.979 + 0.199i$
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  + (−0.250 − 1.98i)3-s + (0.187 − 0.982i)4-s + (0.809 − 0.587i)5-s + (−2.89 + 0.742i)9-s + (0.637 + 0.770i)11-s + (−1.99 − 0.125i)12-s + (−1.36 − 1.45i)15-s + (−0.929 − 0.368i)16-s + (−0.425 − 0.904i)20-s + (−0.211 − 0.134i)23-s + (0.309 − 0.951i)25-s + (1.45 + 3.68i)27-s + (0.200 + 1.05i)31-s + (1.36 − 1.45i)33-s + (0.187 + 2.97i)36-s + (0.700 − 1.76i)37-s + ⋯
L(s)  = 1  + (−0.250 − 1.98i)3-s + (0.187 − 0.982i)4-s + (0.809 − 0.587i)5-s + (−2.89 + 0.742i)9-s + (0.637 + 0.770i)11-s + (−1.99 − 0.125i)12-s + (−1.36 − 1.45i)15-s + (−0.929 − 0.368i)16-s + (−0.425 − 0.904i)20-s + (−0.211 − 0.134i)23-s + (0.309 − 0.951i)25-s + (1.45 + 3.68i)27-s + (0.200 + 1.05i)31-s + (1.36 − 1.45i)33-s + (0.187 + 2.97i)36-s + (0.700 − 1.76i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.979 + 0.199i$
Analytic conductor: \(0.686214\)
Root analytic conductor: \(0.828380\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1044, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ -0.979 + 0.199i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.115236383\)
\(L(\frac12)\) \(\approx\) \(1.115236383\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-0.637 - 0.770i)T \)
good2 \( 1 + (-0.187 + 0.982i)T^{2} \)
3 \( 1 + (0.250 + 1.98i)T + (-0.968 + 0.248i)T^{2} \)
7 \( 1 + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.876 - 0.481i)T^{2} \)
17 \( 1 + (-0.929 + 0.368i)T^{2} \)
19 \( 1 + (-0.968 - 0.248i)T^{2} \)
23 \( 1 + (0.211 + 0.134i)T + (0.425 + 0.904i)T^{2} \)
29 \( 1 + (0.637 - 0.770i)T^{2} \)
31 \( 1 + (-0.200 - 1.05i)T + (-0.929 + 0.368i)T^{2} \)
37 \( 1 + (-0.700 + 1.76i)T + (-0.728 - 0.684i)T^{2} \)
41 \( 1 + (0.425 - 0.904i)T^{2} \)
43 \( 1 + (-0.809 - 0.587i)T^{2} \)
47 \( 1 + (-0.566 + 1.03i)T + (-0.535 - 0.844i)T^{2} \)
53 \( 1 + (-1.05 - 1.12i)T + (-0.0627 + 0.998i)T^{2} \)
59 \( 1 + (0.116 - 1.85i)T + (-0.992 - 0.125i)T^{2} \)
61 \( 1 + (0.425 + 0.904i)T^{2} \)
67 \( 1 + (1.23 + 0.582i)T + (0.637 + 0.770i)T^{2} \)
71 \( 1 + (-0.110 - 0.0604i)T + (0.535 + 0.844i)T^{2} \)
73 \( 1 + (-0.992 + 0.125i)T^{2} \)
79 \( 1 + (-0.968 + 0.248i)T^{2} \)
83 \( 1 + (0.968 + 0.248i)T^{2} \)
89 \( 1 + (0.110 + 1.74i)T + (-0.992 + 0.125i)T^{2} \)
97 \( 1 + (-0.666 + 0.313i)T + (0.637 - 0.770i)T^{2} \)
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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

−9.166627438060949647205098124296, −8.700741243427196717259229330179, −7.45154845135965549666887483913, −6.94570931154707948976989192901, −6.01688576679113643234011739448, −5.72052595027645768844687430036, −4.63789925306525852867635742488, −2.55892437986466957633090528441, −1.79505804774312712030162403029, −1.01026032627064722823017040234, 2.58097615582969730109180651720, 3.36578573506638734976990805046, 4.04229444907931849069853195595, 5.02091571157560271512615123660, 5.99457701494071596106146334637, 6.59244144891468073873867970554, 8.044177905493477852891529953123, 8.753209496791554431977480034977, 9.522350721219286690583543824142, 10.01576497519306638741807929065

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