Properties

Label 2-392-392.221-c1-0-46
Degree $2$
Conductor $392$
Sign $0.777 + 0.628i$
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  + (1.40 − 0.146i)2-s + (0.569 − 0.223i)3-s + (1.95 − 0.411i)4-s + (0.502 − 3.33i)5-s + (0.768 − 0.397i)6-s + (1.43 + 2.22i)7-s + (2.69 − 0.864i)8-s + (−1.92 + 1.78i)9-s + (0.219 − 4.76i)10-s + (−0.283 + 0.305i)11-s + (1.02 − 0.671i)12-s + (−4.40 − 1.00i)13-s + (2.34 + 2.91i)14-s + (−0.459 − 2.01i)15-s + (3.66 − 1.60i)16-s + (−0.0571 + 0.762i)17-s + ⋯
L(s)  = 1  + (0.994 − 0.103i)2-s + (0.328 − 0.129i)3-s + (0.978 − 0.205i)4-s + (0.224 − 1.49i)5-s + (0.313 − 0.162i)6-s + (0.542 + 0.840i)7-s + (0.952 − 0.305i)8-s + (−0.641 + 0.595i)9-s + (0.0694 − 1.50i)10-s + (−0.0853 + 0.0919i)11-s + (0.295 − 0.193i)12-s + (−1.22 − 0.278i)13-s + (0.626 + 0.779i)14-s + (−0.118 − 0.519i)15-s + (0.915 − 0.402i)16-s + (−0.0138 + 0.184i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 + 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.777 + 0.628i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ 0.777 + 0.628i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.60785 - 0.922079i\)
\(L(\frac12)\) \(\approx\) \(2.60785 - 0.922079i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.146i)T \)
7 \( 1 + (-1.43 - 2.22i)T \)
good3 \( 1 + (-0.569 + 0.223i)T + (2.19 - 2.04i)T^{2} \)
5 \( 1 + (-0.502 + 3.33i)T + (-4.77 - 1.47i)T^{2} \)
11 \( 1 + (0.283 - 0.305i)T + (-0.822 - 10.9i)T^{2} \)
13 \( 1 + (4.40 + 1.00i)T + (11.7 + 5.64i)T^{2} \)
17 \( 1 + (0.0571 - 0.762i)T + (-16.8 - 2.53i)T^{2} \)
19 \( 1 + (0.228 - 0.131i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0580 - 0.774i)T + (-22.7 + 3.42i)T^{2} \)
29 \( 1 + (-2.22 - 4.62i)T + (-18.0 + 22.6i)T^{2} \)
31 \( 1 + (-3.35 + 5.80i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.158 - 0.232i)T + (-13.5 - 34.4i)T^{2} \)
41 \( 1 + (4.82 - 6.04i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (6.83 - 5.45i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (-6.65 + 2.05i)T + (38.8 - 26.4i)T^{2} \)
53 \( 1 + (-4.26 - 6.25i)T + (-19.3 + 49.3i)T^{2} \)
59 \( 1 + (-0.176 - 1.17i)T + (-56.3 + 17.3i)T^{2} \)
61 \( 1 + (-5.72 + 8.38i)T + (-22.2 - 56.7i)T^{2} \)
67 \( 1 + (11.7 + 6.77i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.50 + 2.65i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (13.8 + 4.27i)T + (60.3 + 41.1i)T^{2} \)
79 \( 1 + (2.22 + 3.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.7 + 2.44i)T + (74.7 - 36.0i)T^{2} \)
89 \( 1 + (-6.09 + 5.65i)T + (6.65 - 88.7i)T^{2} \)
97 \( 1 - 1.01T + 97T^{2} \)
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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

−11.69107370070411989307550756903, −10.41531990847997163800343582062, −9.282864659156275360399686249528, −8.343032217395648971691837840128, −7.60701969211633784425393855582, −6.00507677280154252555968630674, −5.10897435916820381506979355124, −4.66325127891899760225683366366, −2.83471672634808801496996160189, −1.75181015970706983315603913509, 2.35595443787231998311322850349, 3.24597741801094031122147164551, 4.34167923293393121384653407480, 5.61426454182077826328422149650, 6.82187721254636832419646650203, 7.20373494869964037412422122472, 8.411802264477017780138081021251, 10.02958359739999812179880712100, 10.53130632592629424831727920994, 11.56283799999492197238704871196

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