Properties

Label 2-1380-1380.839-c0-0-4
Degree $2$
Conductor $1380$
Sign $0.938 - 0.343i$
Analytic cond. $0.688709$
Root an. cond. $0.829885$
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  + (0.540 + 0.841i)2-s + (−0.755 − 0.654i)3-s + (−0.415 + 0.909i)4-s + (0.959 − 0.281i)5-s + (0.142 − 0.989i)6-s + (0.989 − 0.857i)7-s + (−0.989 + 0.142i)8-s + (0.142 + 0.989i)9-s + (0.755 + 0.654i)10-s + (0.909 − 0.415i)12-s + (1.25 + 0.368i)14-s + (−0.909 − 0.415i)15-s + (−0.654 − 0.755i)16-s + (−0.755 + 0.654i)18-s + (−0.142 + 0.989i)20-s − 1.30·21-s + ⋯
L(s)  = 1  + (0.540 + 0.841i)2-s + (−0.755 − 0.654i)3-s + (−0.415 + 0.909i)4-s + (0.959 − 0.281i)5-s + (0.142 − 0.989i)6-s + (0.989 − 0.857i)7-s + (−0.989 + 0.142i)8-s + (0.142 + 0.989i)9-s + (0.755 + 0.654i)10-s + (0.909 − 0.415i)12-s + (1.25 + 0.368i)14-s + (−0.909 − 0.415i)15-s + (−0.654 − 0.755i)16-s + (−0.755 + 0.654i)18-s + (−0.142 + 0.989i)20-s − 1.30·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.938 - 0.343i$
Analytic conductor: \(0.688709\)
Root analytic conductor: \(0.829885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (839, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :0),\ 0.938 - 0.343i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.355808652\)
\(L(\frac12)\) \(\approx\) \(1.355808652\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.540 - 0.841i)T \)
3 \( 1 + (0.755 + 0.654i)T \)
5 \( 1 + (-0.959 + 0.281i)T \)
23 \( 1 + (0.540 + 0.841i)T \)
good7 \( 1 + (-0.989 + 0.857i)T + (0.142 - 0.989i)T^{2} \)
11 \( 1 + (-0.415 - 0.909i)T^{2} \)
13 \( 1 + (0.142 + 0.989i)T^{2} \)
17 \( 1 + (0.654 - 0.755i)T^{2} \)
19 \( 1 + (-0.654 - 0.755i)T^{2} \)
29 \( 1 + (-0.512 + 0.234i)T + (0.654 - 0.755i)T^{2} \)
31 \( 1 + (0.959 - 0.281i)T^{2} \)
37 \( 1 + (0.841 + 0.540i)T^{2} \)
41 \( 1 + (-0.425 - 1.45i)T + (-0.841 + 0.540i)T^{2} \)
43 \( 1 + (-0.822 - 0.118i)T + (0.959 + 0.281i)T^{2} \)
47 \( 1 - 1.91iT - T^{2} \)
53 \( 1 + (0.142 - 0.989i)T^{2} \)
59 \( 1 + (-0.142 - 0.989i)T^{2} \)
61 \( 1 + (1.80 - 0.258i)T + (0.959 - 0.281i)T^{2} \)
67 \( 1 + (0.153 + 0.239i)T + (-0.415 + 0.909i)T^{2} \)
71 \( 1 + (0.415 - 0.909i)T^{2} \)
73 \( 1 + (0.654 + 0.755i)T^{2} \)
79 \( 1 + (-0.142 - 0.989i)T^{2} \)
83 \( 1 + (1.89 + 0.557i)T + (0.841 + 0.540i)T^{2} \)
89 \( 1 + (-0.239 + 1.66i)T + (-0.959 - 0.281i)T^{2} \)
97 \( 1 + (0.841 - 0.540i)T^{2} \)
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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

−9.829319572403538954089719105340, −8.725386290269144565307782249387, −7.901514298567160905356863894017, −7.34221347372059315765764716166, −6.29659626532875994926913911596, −5.92189957945883621384034416223, −4.74114313777862577762221068563, −4.47554509475346937654177969641, −2.65122092348204690053227880031, −1.29752381764834689714350010580, 1.53217134780364078173070908754, 2.53867386021650809110801921341, 3.74480944697032341135449170993, 4.78728896597494815706438001293, 5.50935923210155711916895428022, 5.89130614980937521180754022403, 6.97879698495905737197829550239, 8.525860216285868547225720271886, 9.250314303145792431324526581125, 9.901510587358744141552536830328

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