Properties

Label 2-38e2-76.11-c0-0-0
Degree $2$
Conductor $1444$
Sign $0.260 - 0.965i$
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 + (−0.309 + 0.535i)5-s + 0.999·8-s + (−0.5 − 0.866i)9-s + (−0.309 − 0.535i)10-s + (0.809 + 1.40i)13-s + (−0.5 + 0.866i)16-s + (0.809 − 1.40i)17-s + 0.999·18-s + 0.618·20-s + (0.309 + 0.535i)25-s − 1.61·26-s + (0.809 + 1.40i)29-s + (−0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.309 + 0.535i)5-s + 0.999·8-s + (−0.5 − 0.866i)9-s + (−0.309 − 0.535i)10-s + (0.809 + 1.40i)13-s + (−0.5 + 0.866i)16-s + (0.809 − 1.40i)17-s + 0.999·18-s + 0.618·20-s + (0.309 + 0.535i)25-s − 1.61·26-s + (0.809 + 1.40i)29-s + (−0.499 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $0.260 - 0.965i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ 0.260 - 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7778113845\)
\(L(\frac12)\) \(\approx\) \(0.7778113845\)
\(L(1)\) 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 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 0.618T + T^{2} \)
41 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.590874756124860292172783993196, −9.024582763766869555240995995536, −8.376698530956970110009554601560, −7.21144978942078413117554758733, −6.86133161171460556272805727205, −6.00947447894086019283366793857, −5.08320619392026124283305820207, −4.01244822827102039519638721661, −2.96374106380992548475863701087, −1.21284222050785736274944464138, 0.959189799220202561948281783597, 2.33048224650866600996309818980, 3.37025536524545702042987333773, 4.26215106689646145851834617072, 5.27825611399082580354943955214, 6.15264927609710472273709616419, 7.67333997498007329163238145696, 8.228500569589898615409102311765, 8.516431679892397278910482777882, 9.695086900974791720222496119532

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