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Properties

Label 2-38e2-19.18-c2-0-39
Degree $2$
Conductor $1444$
Sign $0.989 + 0.143i$
Analytic cond. $39.3461$
Root an. cond. $6.27265$
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  + 0.650i·3-s + 4.27·5-s + 1.20·7-s + 8.57·9-s + 14.1·11-s − 9.56i·13-s + 2.77i·15-s + 11.1·17-s + 0.780i·21-s + 20.6·23-s − 6.72·25-s + 11.4i·27-s − 17.7i·29-s − 35.5i·31-s + 9.20i·33-s + ⋯
L(s)  = 1  + 0.216i·3-s + 0.855·5-s + 0.171·7-s + 0.953·9-s + 1.28·11-s − 0.735i·13-s + 0.185i·15-s + 0.655·17-s + 0.0371i·21-s + 0.896·23-s − 0.268·25-s + 0.423i·27-s − 0.610i·29-s − 1.14i·31-s + 0.278i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $0.989 + 0.143i$
Analytic conductor: \(39.3461\)
Root analytic conductor: \(6.27265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :1),\ 0.989 + 0.143i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.997789794\)
\(L(\frac12)\) \(\approx\) \(2.997789794\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 - 0.650iT - 9T^{2} \)
5 \( 1 - 4.27T + 25T^{2} \)
7 \( 1 - 1.20T + 49T^{2} \)
11 \( 1 - 14.1T + 121T^{2} \)
13 \( 1 + 9.56iT - 169T^{2} \)
17 \( 1 - 11.1T + 289T^{2} \)
23 \( 1 - 20.6T + 529T^{2} \)
29 \( 1 + 17.7iT - 841T^{2} \)
31 \( 1 + 35.5iT - 961T^{2} \)
37 \( 1 - 46.5iT - 1.36e3T^{2} \)
41 \( 1 + 47.5iT - 1.68e3T^{2} \)
43 \( 1 + 41.3T + 1.84e3T^{2} \)
47 \( 1 - 16.1T + 2.20e3T^{2} \)
53 \( 1 + 42.1iT - 2.80e3T^{2} \)
59 \( 1 - 60.3iT - 3.48e3T^{2} \)
61 \( 1 - 8.70T + 3.72e3T^{2} \)
67 \( 1 + 88.9iT - 4.48e3T^{2} \)
71 \( 1 - 104. iT - 5.04e3T^{2} \)
73 \( 1 + 112.T + 5.32e3T^{2} \)
79 \( 1 - 15.7iT - 6.24e3T^{2} \)
83 \( 1 - 145.T + 6.88e3T^{2} \)
89 \( 1 + 59.2iT - 7.92e3T^{2} \)
97 \( 1 - 173. iT - 9.40e3T^{2} \)
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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

−9.597426302508333473450044550126, −8.627957714120046084865488250658, −7.67912930979936093672646968354, −6.81123103321925714376305519544, −6.04110134665225405481158052179, −5.17916383681570685371845219434, −4.22285265016833167556290463434, −3.29754520795505226752669826473, −1.94741083221251485643698129931, −1.00827801665051087302314938202, 1.24627413206407828748930623636, 1.82526301112470433071319476121, 3.28724436613198614123300768676, 4.30727663484591599592395987095, 5.18257254525469808051675891794, 6.27032619852672072003962448857, 6.81311767375135982800030969581, 7.59880061068459883547599759622, 8.770277621523952894027096263492, 9.380261349577861230751316223461

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