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Properties

Label 2-392-8.3-c2-0-9
Degree $2$
Conductor $392$
Sign $0.883 - 0.467i$
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  + (0.707 − 1.87i)2-s − 3.41·3-s + (−3 − 2.64i)4-s − 1.54i·5-s + (−2.41 + 6.38i)6-s + (−7.07 + 3.74i)8-s + 2.65·9-s + (−2.89 − 1.09i)10-s − 4.48·11-s + (10.2 + 9.03i)12-s + 1.54i·13-s + 5.29i·15-s + (1.99 + 15.8i)16-s − 23.6·17-s + (1.87 − 4.97i)18-s + 24.8·19-s + ⋯
L(s)  = 1  + (0.353 − 0.935i)2-s − 1.13·3-s + (−0.750 − 0.661i)4-s − 0.309i·5-s + (−0.402 + 1.06i)6-s + (−0.883 + 0.467i)8-s + 0.295·9-s + (−0.289 − 0.109i)10-s − 0.407·11-s + (0.853 + 0.752i)12-s + 0.119i·13-s + 0.352i·15-s + (0.124 + 0.992i)16-s − 1.39·17-s + (0.104 − 0.276i)18-s + 1.30·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.467i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.883 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.883 - 0.467i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ 0.883 - 0.467i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.538556 + 0.133706i\)
\(L(\frac12)\) \(\approx\) \(0.538556 + 0.133706i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.87i)T \)
7 \( 1 \)
good3 \( 1 + 3.41T + 9T^{2} \)
5 \( 1 + 1.54iT - 25T^{2} \)
11 \( 1 + 4.48T + 121T^{2} \)
13 \( 1 - 1.54iT - 169T^{2} \)
17 \( 1 + 23.6T + 289T^{2} \)
19 \( 1 - 24.8T + 361T^{2} \)
23 \( 1 - 35.2iT - 529T^{2} \)
29 \( 1 + 22.4iT - 841T^{2} \)
31 \( 1 - 46.7iT - 961T^{2} \)
37 \( 1 + 58.5iT - 1.36e3T^{2} \)
41 \( 1 - 26.9T + 1.68e3T^{2} \)
43 \( 1 + 17.1T + 1.84e3T^{2} \)
47 \( 1 - 36.1iT - 2.20e3T^{2} \)
53 \( 1 - 97.8iT - 2.80e3T^{2} \)
59 \( 1 + 61.5T + 3.48e3T^{2} \)
61 \( 1 - 37.6iT - 3.72e3T^{2} \)
67 \( 1 + 33.3T + 4.48e3T^{2} \)
71 \( 1 - 102. iT - 5.04e3T^{2} \)
73 \( 1 + 69.3T + 5.32e3T^{2} \)
79 \( 1 - 38.7iT - 6.24e3T^{2} \)
83 \( 1 + 3.61T + 6.88e3T^{2} \)
89 \( 1 + 44.0T + 7.92e3T^{2} \)
97 \( 1 + 96.1T + 9.40e3T^{2} \)
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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

−11.17343532412179966271547087975, −10.67672175319556813086213468885, −9.534207185178368513879527033195, −8.756139857429187211541511786370, −7.25508406428656999142708577909, −5.95120997272303325258393889002, −5.25207907115625649029307415187, −4.34903716733845480720026991196, −2.87305480223995085681210261470, −1.19841843532453735226234444236, 0.28586478318853999142663421215, 2.95239874201956033519604451474, 4.54509872605022486950747829215, 5.25004983354122128759936211904, 6.32978337696273549440693889327, 6.86653718514355698923772941380, 8.048591557359130644660604064501, 9.022533928668572947731823439139, 10.21739887206246640514926676974, 11.15333738054153898383270494413

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