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Properties

Label 2-392-56.19-c1-0-5
Degree $2$
Conductor $392$
Sign $0.580 - 0.814i$
Analytic cond. $3.13013$
Root an. cond. $1.76921$
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  + (−0.707 − 1.22i)2-s + (−0.937 + 0.541i)3-s + (−0.999 + 1.73i)4-s + (0.662 − 1.14i)5-s + (1.32 + 0.765i)6-s + 2.82·8-s + (−0.914 + 1.58i)9-s − 1.87·10-s + (1 + 1.73i)11-s − 2.16i·12-s − 5.85·13-s + 1.43i·15-s + (−2.00 − 3.46i)16-s + (−3.86 + 2.23i)17-s + 2.58·18-s + (3.58 + 2.07i)19-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)2-s + (−0.541 + 0.312i)3-s + (−0.499 + 0.866i)4-s + (0.296 − 0.513i)5-s + (0.541 + 0.312i)6-s + 0.999·8-s + (−0.304 + 0.527i)9-s − 0.592·10-s + (0.301 + 0.522i)11-s − 0.624i·12-s − 1.62·13-s + 0.370i·15-s + (−0.500 − 0.866i)16-s + (−0.936 + 0.540i)17-s + 0.609·18-s + (0.823 + 0.475i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.580 - 0.814i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ 0.580 - 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.540531 + 0.278451i\)
\(L(\frac12)\) \(\approx\) \(0.540531 + 0.278451i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.22i)T \)
7 \( 1 \)
good3 \( 1 + (0.937 - 0.541i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.662 + 1.14i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.85T + 13T^{2} \)
17 \( 1 + (3.86 - 2.23i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.58 - 2.07i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.24 - 4.18i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.47iT - 29T^{2} \)
31 \( 1 + (-3.20 - 5.54i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.12 + 1.22i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.317iT - 41T^{2} \)
43 \( 1 + 3.17T + 43T^{2} \)
47 \( 1 + (6.40 - 11.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 - 1.73i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.937 - 0.541i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.80 + 8.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-6.12 + 3.53i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.75 - 1.01i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.67iT - 83T^{2} \)
89 \( 1 + (11.0 + 6.37i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.23iT - 97T^{2} \)
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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

−11.30861090793544080040441313625, −10.59774726274171689556595130442, −9.645847735720077567133108712994, −9.075573397929948425154297662504, −7.893368407710514401996533018573, −6.92456925611846108728504423270, −5.13885795630265937721859480868, −4.73846363882754766337796961162, −3.07058775315708753709272116459, −1.63764318702709672350522757386, 0.51352975735602340360992150858, 2.66886328101910351944450285616, 4.65486833401976767578009016432, 5.58911188112068512624391147059, 6.79452639647464664321256250470, 6.94885328241975888245318590775, 8.377218182833869003968340988732, 9.321016735617053918300075479981, 10.04782532548197344977726305619, 11.15467609281663647669317373192

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