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Properties

Label 2-38e2-76.35-c0-0-3
Degree $2$
Conductor $1444$
Sign $-0.516 + 0.855i$
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  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.107 − 0.608i)5-s + (0.500 − 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.473 + 0.397i)10-s + (−1.52 − 0.553i)13-s + (−0.939 + 0.342i)16-s + (−1.23 − 1.04i)17-s − 18-s + 0.618·20-s + (0.580 + 0.211i)25-s + (0.809 + 1.40i)26-s + (1.23 − 1.04i)29-s + (0.939 + 0.342i)32-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.107 − 0.608i)5-s + (0.500 − 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.473 + 0.397i)10-s + (−1.52 − 0.553i)13-s + (−0.939 + 0.342i)16-s + (−1.23 − 1.04i)17-s − 18-s + 0.618·20-s + (0.580 + 0.211i)25-s + (0.809 + 1.40i)26-s + (1.23 − 1.04i)29-s + (0.939 + 0.342i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.516 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.516 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $-0.516 + 0.855i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ -0.516 + 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6613332164\)
\(L(\frac12)\) \(\approx\) \(0.6613332164\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 \)
good3 \( 1 + (-0.766 + 0.642i)T^{2} \)
5 \( 1 + (-0.107 + 0.608i)T + (-0.939 - 0.342i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (1.52 + 0.553i)T + (0.766 + 0.642i)T^{2} \)
17 \( 1 + (1.23 + 1.04i)T + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-1.23 + 1.04i)T + (0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 0.618T + T^{2} \)
41 \( 1 + (-0.580 + 0.211i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (0.107 + 0.608i)T + (-0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.280 + 1.59i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (0.580 - 0.211i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.580 - 0.211i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (-1.23 - 1.04i)T + (0.173 + 0.984i)T^{2} \)
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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

−9.500379230846247765824911016778, −8.910548166667318402134759844970, −7.956499557200548211799090607141, −7.18826322762593171339743280878, −6.49695617515390954327070624666, −4.93742045282432915502773338833, −4.39736040158886790752304565733, −3.08025006432274877124035186008, −2.13603889831888095821234411746, −0.69204875077048888336504983668, 1.73003146783045559097353732961, 2.63903671755611570237179309014, 4.40287081048288851626454129034, 4.99710182411902698069255722404, 6.23797608201826905848287410797, 6.94890826422431482749474890402, 7.38755658001048625342627407392, 8.417991982779142788307883369888, 9.091997795651890885449314301471, 10.11355714638869895704982062323

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