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Properties

Label 2-1375-1375.1121-c0-0-0
Degree $2$
Conductor $1375$
Sign $0.910 - 0.414i$
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  + (−0.456 + 0.969i)3-s + (−0.992 − 0.125i)4-s + (−0.809 − 0.587i)5-s + (−0.0946 − 0.114i)9-s + (0.0627 − 0.998i)11-s + (0.574 − 0.904i)12-s + (0.939 − 0.516i)15-s + (0.968 + 0.248i)16-s + (0.728 + 0.684i)20-s + (0.791 + 0.313i)23-s + (0.309 + 0.951i)25-s + (−0.883 + 0.226i)27-s + (1.84 − 0.233i)31-s + (0.939 + 0.516i)33-s + (0.0795 + 0.125i)36-s + (0.598 + 0.153i)37-s + ⋯
L(s)  = 1  + (−0.456 + 0.969i)3-s + (−0.992 − 0.125i)4-s + (−0.809 − 0.587i)5-s + (−0.0946 − 0.114i)9-s + (0.0627 − 0.998i)11-s + (0.574 − 0.904i)12-s + (0.939 − 0.516i)15-s + (0.968 + 0.248i)16-s + (0.728 + 0.684i)20-s + (0.791 + 0.313i)23-s + (0.309 + 0.951i)25-s + (−0.883 + 0.226i)27-s + (1.84 − 0.233i)31-s + (0.939 + 0.516i)33-s + (0.0795 + 0.125i)36-s + (0.598 + 0.153i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.910 - 0.414i$
Analytic conductor: \(0.686214\)
Root analytic conductor: \(0.828380\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ 0.910 - 0.414i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6318281740\)
\(L(\frac12)\) \(\approx\) \(0.6318281740\)
\(L(1)\) 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 \( 1 + (-0.0627 + 0.998i)T \)
good2 \( 1 + (0.992 + 0.125i)T^{2} \)
3 \( 1 + (0.456 - 0.969i)T + (-0.637 - 0.770i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.187 - 0.982i)T^{2} \)
17 \( 1 + (-0.968 + 0.248i)T^{2} \)
19 \( 1 + (0.637 - 0.770i)T^{2} \)
23 \( 1 + (-0.791 - 0.313i)T + (0.728 + 0.684i)T^{2} \)
29 \( 1 + (-0.0627 - 0.998i)T^{2} \)
31 \( 1 + (-1.84 + 0.233i)T + (0.968 - 0.248i)T^{2} \)
37 \( 1 + (-0.598 - 0.153i)T + (0.876 + 0.481i)T^{2} \)
41 \( 1 + (-0.728 + 0.684i)T^{2} \)
43 \( 1 + (0.809 - 0.587i)T^{2} \)
47 \( 1 + (-0.303 - 1.58i)T + (-0.929 + 0.368i)T^{2} \)
53 \( 1 + (-0.110 + 0.0604i)T + (0.535 - 0.844i)T^{2} \)
59 \( 1 + (-1.03 + 1.63i)T + (-0.425 - 0.904i)T^{2} \)
61 \( 1 + (-0.728 - 0.684i)T^{2} \)
67 \( 1 + (-1.27 + 1.19i)T + (0.0627 - 0.998i)T^{2} \)
71 \( 1 + (0.200 + 1.05i)T + (-0.929 + 0.368i)T^{2} \)
73 \( 1 + (0.425 - 0.904i)T^{2} \)
79 \( 1 + (0.637 + 0.770i)T^{2} \)
83 \( 1 + (0.637 - 0.770i)T^{2} \)
89 \( 1 + (0.200 + 0.316i)T + (-0.425 + 0.904i)T^{2} \)
97 \( 1 + (-1.41 - 1.32i)T + (0.0627 + 0.998i)T^{2} \)
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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

−9.660436384441785472604881858002, −9.195160446170170811245517435268, −8.298931877222993406411843875610, −7.75903006997032694972031703353, −6.30386408539120444743004359308, −5.34251759042631937597584394062, −4.72232518571759531360420896308, −4.07368216958648300633299267268, −3.20042526151860866397932749727, −0.898481222263506974973620214188, 0.906118023088098211926350677781, 2.54543037234698908615898295221, 3.83329071737184601342181131701, 4.55512036675343031231109471580, 5.59784131982563241367079803012, 6.77380226871262647052958181946, 7.14014720083705049636380424615, 8.053129095190275373075472868179, 8.719260187154266079594827496868, 9.837457655609267487581487493675

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