Properties

Label 2-38e2-76.7-c0-0-1
Degree $2$
Conductor $1444$
Sign $0.636 - 0.771i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.809 + 1.40i)5-s + 0.999·8-s + (−0.5 + 0.866i)9-s + (0.809 − 1.40i)10-s + (−0.309 + 0.535i)13-s + (−0.5 − 0.866i)16-s + (−0.309 − 0.535i)17-s + 0.999·18-s − 1.61·20-s + (−0.809 + 1.40i)25-s + 0.618·26-s + (−0.309 + 0.535i)29-s + (−0.499 + 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.809 + 1.40i)5-s + 0.999·8-s + (−0.5 + 0.866i)9-s + (0.809 − 1.40i)10-s + (−0.309 + 0.535i)13-s + (−0.5 − 0.866i)16-s + (−0.309 − 0.535i)17-s + 0.999·18-s − 1.61·20-s + (−0.809 + 1.40i)25-s + 0.618·26-s + (−0.309 + 0.535i)29-s + (−0.499 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $0.636 - 0.771i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (1375, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ 0.636 - 0.771i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8075912998\)
\(L(\frac12)\) \(\approx\) \(0.8075912998\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 \)
good3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 1.61T + T^{2} \)
41 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.03848423497485658477556538856, −9.225132979344759879161823411930, −8.422872852838874027877214753024, −7.36334336426060933337947603882, −6.86292690294591347329559660141, −5.68921742718386917897181156451, −4.70422006356188020422658812839, −3.42203369604030025271906319742, −2.57999526727318222766026317581, −1.90919156669832866666906432131, 0.78127675763070228642618771759, 2.06607959945994590687584689633, 3.86314930709287956649059214821, 4.90999343305934473845420631102, 5.64709729268612823645904710603, 6.15819531598493370851108556594, 7.23194614467255245420540359089, 8.193992782758745333592010595344, 8.937916788717833124574499501562, 9.215901875194306133031509945630

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