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Properties

Label 2-1380-5.4-c1-0-7
Degree $2$
Conductor $1380$
Sign $0.997 + 0.0744i$
Analytic cond. $11.0193$
Root an. cond. $3.31954$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s + (−2.22 − 0.166i)5-s + 1.74i·7-s − 9-s + 2.39·11-s − 3.50i·13-s + (−0.166 + 2.22i)15-s + 5.88i·17-s − 1.93·19-s + 1.74·21-s + i·23-s + (4.94 + 0.742i)25-s + i·27-s + 4.22·29-s + 1.61·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.997 − 0.0744i)5-s + 0.658i·7-s − 0.333·9-s + 0.723·11-s − 0.971i·13-s + (−0.0429 + 0.575i)15-s + 1.42i·17-s − 0.443·19-s + 0.380·21-s + 0.208i·23-s + (0.988 + 0.148i)25-s + 0.192i·27-s + 0.783·29-s + 0.289·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.997 + 0.0744i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ 0.997 + 0.0744i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.338164887\)
\(L(\frac12)\) \(\approx\) \(1.338164887\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 + (2.22 + 0.166i)T \)
23 \( 1 - iT \)
good7 \( 1 - 1.74iT - 7T^{2} \)
11 \( 1 - 2.39T + 11T^{2} \)
13 \( 1 + 3.50iT - 13T^{2} \)
17 \( 1 - 5.88iT - 17T^{2} \)
19 \( 1 + 1.93T + 19T^{2} \)
29 \( 1 - 4.22T + 29T^{2} \)
31 \( 1 - 1.61T + 31T^{2} \)
37 \( 1 + 4.06iT - 37T^{2} \)
41 \( 1 - 8.46T + 41T^{2} \)
43 \( 1 + 2.70iT - 43T^{2} \)
47 \( 1 - 8.18iT - 47T^{2} \)
53 \( 1 - 2.78iT - 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 - 3.35T + 61T^{2} \)
67 \( 1 + 2.16iT - 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + 4.98iT - 73T^{2} \)
79 \( 1 - 6.25T + 79T^{2} \)
83 \( 1 - 1.55iT - 83T^{2} \)
89 \( 1 + 7.14T + 89T^{2} \)
97 \( 1 + 5.93iT - 97T^{2} \)
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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

−9.394122772297506107077678792143, −8.480263030523093477679476806129, −8.120439896517673865448445485556, −7.19978353280405247877094473344, −6.29332837469649968362563320625, −5.56804969859346636469457677857, −4.34500409178822507674433233343, −3.49558546673148359767175351129, −2.36134649558028080735190656413, −0.930391738826573461053025740222, 0.77578520712244533159294781128, 2.60074209006617557572452429705, 3.79126488891539207787650124989, 4.30706196462470235962667643412, 5.13520872360477265857882795084, 6.60427525889816943263877584767, 7.02219398652581101220550239904, 8.044514810400736379202434698839, 8.827531044470526782234793625946, 9.572379201313939582204630022614

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