Properties

Label 2-1375-1375.439-c0-0-0
Degree $2$
Conductor $1375$
Sign $0.711 - 0.702i$
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  + (0.0623 + 0.242i)3-s + (0.929 + 0.368i)4-s + (−0.309 + 0.951i)5-s + (0.821 − 0.451i)9-s + (0.187 − 0.982i)11-s + (−0.0314 + 0.248i)12-s + (−0.250 − 0.0157i)15-s + (0.728 + 0.684i)16-s + (−0.637 + 0.770i)20-s + (0.450 − 0.211i)23-s + (−0.809 − 0.587i)25-s + (0.332 + 0.353i)27-s + (−0.791 + 0.313i)31-s + (0.250 − 0.0157i)33-s + (0.929 − 0.117i)36-s + (−0.804 + 0.856i)37-s + ⋯
L(s)  = 1  + (0.0623 + 0.242i)3-s + (0.929 + 0.368i)4-s + (−0.309 + 0.951i)5-s + (0.821 − 0.451i)9-s + (0.187 − 0.982i)11-s + (−0.0314 + 0.248i)12-s + (−0.250 − 0.0157i)15-s + (0.728 + 0.684i)16-s + (−0.637 + 0.770i)20-s + (0.450 − 0.211i)23-s + (−0.809 − 0.587i)25-s + (0.332 + 0.353i)27-s + (−0.791 + 0.313i)31-s + (0.250 − 0.0157i)33-s + (0.929 − 0.117i)36-s + (−0.804 + 0.856i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.711 - 0.702i$
Analytic conductor: \(0.686214\)
Root analytic conductor: \(0.828380\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ 0.711 - 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.380076707\)
\(L(\frac12)\) \(\approx\) \(1.380076707\)
\(L(1)\) 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 \( 1 + (-0.187 + 0.982i)T \)
good2 \( 1 + (-0.929 - 0.368i)T^{2} \)
3 \( 1 + (-0.0623 - 0.242i)T + (-0.876 + 0.481i)T^{2} \)
7 \( 1 + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.535 - 0.844i)T^{2} \)
17 \( 1 + (0.728 - 0.684i)T^{2} \)
19 \( 1 + (-0.876 - 0.481i)T^{2} \)
23 \( 1 + (-0.450 + 0.211i)T + (0.637 - 0.770i)T^{2} \)
29 \( 1 + (0.187 + 0.982i)T^{2} \)
31 \( 1 + (0.791 - 0.313i)T + (0.728 - 0.684i)T^{2} \)
37 \( 1 + (0.804 - 0.856i)T + (-0.0627 - 0.998i)T^{2} \)
41 \( 1 + (0.637 + 0.770i)T^{2} \)
43 \( 1 + (0.309 + 0.951i)T^{2} \)
47 \( 1 + (1.60 - 1.01i)T + (0.425 - 0.904i)T^{2} \)
53 \( 1 + (1.96 + 0.123i)T + (0.992 + 0.125i)T^{2} \)
59 \( 1 + (1.44 + 0.182i)T + (0.968 + 0.248i)T^{2} \)
61 \( 1 + (0.637 - 0.770i)T^{2} \)
67 \( 1 + (-1.53 + 1.27i)T + (0.187 - 0.982i)T^{2} \)
71 \( 1 + (1.06 + 1.67i)T + (-0.425 + 0.904i)T^{2} \)
73 \( 1 + (0.968 - 0.248i)T^{2} \)
79 \( 1 + (-0.876 + 0.481i)T^{2} \)
83 \( 1 + (0.876 + 0.481i)T^{2} \)
89 \( 1 + (-1.06 + 0.134i)T + (0.968 - 0.248i)T^{2} \)
97 \( 1 + (-1.05 - 0.872i)T + (0.187 + 0.982i)T^{2} \)
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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

−10.01881556230514110801134494217, −9.111072032654996535389774017776, −8.072121101625249968960827174065, −7.41444472229275651377553352533, −6.56561413138185543853763524157, −6.14903513593182501200365671110, −4.69677962797321896562074378008, −3.43421508522224173423418413389, −3.15381067733965425408927504956, −1.68999365872134201553695082455, 1.40422919559022059353558344525, 2.15289711649007428283095471935, 3.66694186054654710989854524215, 4.73392218894881565892041049713, 5.41597252869686296705449289244, 6.57177233901611216997668348899, 7.31664578125392719028841088283, 7.82669987276126587099961042020, 8.916589137763245703693120591030, 9.749466259463256120623858793686

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