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Properties

Label 2-1375-1375.516-c0-0-0
Degree $2$
Conductor $1375$
Sign $0.448 - 0.893i$
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  + (0.542 + 0.656i)3-s + (0.968 + 0.248i)4-s + (0.309 + 0.951i)5-s + (0.0515 − 0.269i)9-s + (−0.992 − 0.125i)11-s + (0.362 + 0.770i)12-s + (−0.456 + 0.718i)15-s + (0.876 + 0.481i)16-s + (0.0627 + 0.998i)20-s + (−0.929 − 0.872i)23-s + (−0.809 + 0.587i)25-s + (0.951 − 0.522i)27-s + (1.41 − 0.362i)31-s + (−0.456 − 0.718i)33-s + (0.117 − 0.248i)36-s + (−1.41 − 0.779i)37-s + ⋯
L(s)  = 1  + (0.542 + 0.656i)3-s + (0.968 + 0.248i)4-s + (0.309 + 0.951i)5-s + (0.0515 − 0.269i)9-s + (−0.992 − 0.125i)11-s + (0.362 + 0.770i)12-s + (−0.456 + 0.718i)15-s + (0.876 + 0.481i)16-s + (0.0627 + 0.998i)20-s + (−0.929 − 0.872i)23-s + (−0.809 + 0.587i)25-s + (0.951 − 0.522i)27-s + (1.41 − 0.362i)31-s + (−0.456 − 0.718i)33-s + (0.117 − 0.248i)36-s + (−1.41 − 0.779i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.448 - 0.893i$
Analytic conductor: \(0.686214\)
Root analytic conductor: \(0.828380\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (516, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ 0.448 - 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.603342230\)
\(L(\frac12)\) \(\approx\) \(1.603342230\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (0.992 + 0.125i)T \)
good2 \( 1 + (-0.968 - 0.248i)T^{2} \)
3 \( 1 + (-0.542 - 0.656i)T + (-0.187 + 0.982i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.929 + 0.368i)T^{2} \)
17 \( 1 + (-0.876 + 0.481i)T^{2} \)
19 \( 1 + (0.187 + 0.982i)T^{2} \)
23 \( 1 + (0.929 + 0.872i)T + (0.0627 + 0.998i)T^{2} \)
29 \( 1 + (0.992 - 0.125i)T^{2} \)
31 \( 1 + (-1.41 + 0.362i)T + (0.876 - 0.481i)T^{2} \)
37 \( 1 + (1.41 + 0.779i)T + (0.535 + 0.844i)T^{2} \)
41 \( 1 + (-0.0627 + 0.998i)T^{2} \)
43 \( 1 + (-0.309 + 0.951i)T^{2} \)
47 \( 1 + (0.574 - 0.227i)T + (0.728 - 0.684i)T^{2} \)
53 \( 1 + (1.06 - 1.67i)T + (-0.425 - 0.904i)T^{2} \)
59 \( 1 + (0.746 + 1.58i)T + (-0.637 + 0.770i)T^{2} \)
61 \( 1 + (-0.0627 - 0.998i)T^{2} \)
67 \( 1 + (-0.0672 + 1.06i)T + (-0.992 - 0.125i)T^{2} \)
71 \( 1 + (-0.791 + 0.313i)T + (0.728 - 0.684i)T^{2} \)
73 \( 1 + (0.637 + 0.770i)T^{2} \)
79 \( 1 + (0.187 - 0.982i)T^{2} \)
83 \( 1 + (0.187 + 0.982i)T^{2} \)
89 \( 1 + (-0.791 + 1.68i)T + (-0.637 - 0.770i)T^{2} \)
97 \( 1 + (-0.110 - 1.74i)T + (-0.992 + 0.125i)T^{2} \)
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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

−10.14687929349737243972271219545, −9.246021042619522304673141000510, −8.145538905671528860038289730174, −7.61933754642618624548824302283, −6.50671015167695544107503843952, −6.11964567214697439021722333394, −4.77618180725886782506625995222, −3.56278989398208822373116171711, −2.92569531630095994240828212074, −2.08619148406347412890827850148, 1.47188784177027240342647154405, 2.20348861378972512932701772302, 3.23068229612603698777377824179, 4.79083622601090308267121274572, 5.47605129826778314227917329444, 6.44736008200831344076734421103, 7.30547454218038929498186547952, 8.098090027220845182478729616195, 8.483494219007346031422584877843, 9.844233907951156048918128983272

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