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Properties

Label 2-38e2-19.13-c0-0-2
Degree $2$
Conductor $1444$
Sign $0.756 + 0.654i$
Analytic cond. $0.720649$
Root an. cond. $0.848910$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 − 0.984i)5-s + (0.5 − 0.866i)7-s + (0.766 + 0.642i)9-s + (0.5 + 0.866i)11-s + (−0.766 + 0.642i)17-s + (0.347 − 1.96i)23-s + (−0.939 − 0.342i)35-s + (−0.173 − 0.984i)43-s + (0.5 − 0.866i)45-s + (−0.766 − 0.642i)47-s + (0.766 − 0.642i)55-s + (−0.173 + 0.984i)61-s + (0.939 − 0.342i)63-s + (0.939 + 0.342i)73-s + 0.999·77-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)5-s + (0.5 − 0.866i)7-s + (0.766 + 0.642i)9-s + (0.5 + 0.866i)11-s + (−0.766 + 0.642i)17-s + (0.347 − 1.96i)23-s + (−0.939 − 0.342i)35-s + (−0.173 − 0.984i)43-s + (0.5 − 0.866i)45-s + (−0.766 − 0.642i)47-s + (0.766 − 0.642i)55-s + (−0.173 + 0.984i)61-s + (0.939 − 0.342i)63-s + (0.939 + 0.342i)73-s + 0.999·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $0.756 + 0.654i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ 0.756 + 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.193852765\)
\(L(\frac12)\) \(\approx\) \(1.193852765\)
\(L(1)\) 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 \( 1 + (-0.766 - 0.642i)T^{2} \)
5 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
23 \( 1 + (-0.347 + 1.96i)T + (-0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
53 \( 1 + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.766 + 0.642i)T^{2} \)
97 \( 1 + (-0.173 + 0.984i)T^{2} \)
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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

−9.657578550575018497446384065862, −8.667460430270846566343263709993, −8.166662536804208817896250103342, −7.15939018981629681069160374730, −6.61857664787464691611245880116, −5.16782434056809346188552814760, −4.44584563969055642202848499214, −4.08591757090576855183888262331, −2.22966226946056225556464826614, −1.18261387442914511522637451506, 1.56033483914506031062019341689, 2.90531903490272118231621679504, 3.63255962710652532149526077022, 4.79951486384512001158566651136, 5.81411447069883320557557037233, 6.61480820465365939561459036228, 7.28416328862524800532997119226, 8.201256778679885632052242881099, 9.177587320676378765366627570587, 9.602402202595463281478134415015

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