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Properties

Label 2-392-56.51-c2-0-1
Degree $2$
Conductor $392$
Sign $-0.0819 - 0.996i$
Analytic cond. $10.6812$
Root an. cond. $3.26821$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.575 + 1.91i)2-s + (−2.61 − 4.52i)3-s + (−3.33 + 2.20i)4-s + (−5.42 − 3.13i)5-s + (7.16 − 7.60i)6-s + (−6.14 − 5.12i)8-s + (−9.14 + 15.8i)9-s + (2.87 − 12.2i)10-s + (−4.90 − 8.49i)11-s + (18.6 + 9.33i)12-s + 2.41i·13-s + 32.7i·15-s + (6.27 − 14.7i)16-s + (3.44 + 5.97i)17-s + (−35.5 − 8.39i)18-s + (1.38 − 2.40i)19-s + ⋯
L(s)  = 1  + (0.287 + 0.957i)2-s + (−0.870 − 1.50i)3-s + (−0.834 + 0.551i)4-s + (−1.08 − 0.626i)5-s + (1.19 − 1.26i)6-s + (−0.768 − 0.640i)8-s + (−1.01 + 1.75i)9-s + (0.287 − 1.22i)10-s + (−0.445 − 0.772i)11-s + (1.55 + 0.778i)12-s + 0.185i·13-s + 2.18i·15-s + (0.392 − 0.919i)16-s + (0.202 + 0.351i)17-s + (−1.97 − 0.466i)18-s + (0.0730 − 0.126i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0819 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0819 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.0819 - 0.996i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ -0.0819 - 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.275598 + 0.299198i\)
\(L(\frac12)\) \(\approx\) \(0.275598 + 0.299198i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.575 - 1.91i)T \)
7 \( 1 \)
good3 \( 1 + (2.61 + 4.52i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (5.42 + 3.13i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (4.90 + 8.49i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 2.41iT - 169T^{2} \)
17 \( 1 + (-3.44 - 5.97i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-1.38 + 2.40i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-37.0 - 21.4i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 37.3iT - 841T^{2} \)
31 \( 1 + (6.20 - 3.58i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (0.175 + 0.101i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 63.5T + 1.68e3T^{2} \)
43 \( 1 + 35.3T + 1.84e3T^{2} \)
47 \( 1 + (-32.8 - 18.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-47.3 + 27.3i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-52.3 - 90.7i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (37.9 + 21.8i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (15.5 + 26.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 23.1iT - 5.04e3T^{2} \)
73 \( 1 + (34.6 + 59.9i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-17.2 - 9.96i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 5.11T + 6.88e3T^{2} \)
89 \( 1 + (8.99 - 15.5i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 12.4T + 9.40e3T^{2} \)
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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

−11.76776959287252139395740497525, −10.78269525706988421813507652985, −8.910638777145918699137519713491, −8.212527742685042570125414449827, −7.39984774102138739221132224984, −6.77982753643154507821934516921, −5.61856713327381496004215106327, −4.93629336826273222390419123097, −3.37015067960684709878939858661, −1.01767608208821484958135721309, 0.23517362902267529284147408568, 2.89657492904337933689656031560, 3.87955181091246427123301129987, 4.67937198994939367166212693691, 5.48919357643685352056290592762, 6.91951855613980202217044421418, 8.385944110915874728181181351593, 9.525482613589124992318922210469, 10.26245416182669513524345615237, 10.85058246029912840917325150985

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