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Properties

Label 2-38e2-76.43-c0-0-2
Degree $2$
Conductor $1444$
Sign $0.872 + 0.489i$
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  + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (0.473 + 0.397i)5-s + (0.500 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.580 + 0.211i)10-s + (0.280 + 1.59i)13-s + (0.173 − 0.984i)16-s + (1.52 − 0.553i)17-s − 18-s + 0.618·20-s + (−0.107 − 0.608i)25-s + (0.809 + 1.40i)26-s + (−1.52 − 0.553i)29-s + (−0.173 − 0.984i)32-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (0.473 + 0.397i)5-s + (0.500 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.580 + 0.211i)10-s + (0.280 + 1.59i)13-s + (0.173 − 0.984i)16-s + (1.52 − 0.553i)17-s − 18-s + 0.618·20-s + (−0.107 − 0.608i)25-s + (0.809 + 1.40i)26-s + (−1.52 − 0.553i)29-s + (−0.173 − 0.984i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $0.872 + 0.489i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ 0.872 + 0.489i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.007950949\)
\(L(\frac12)\) \(\approx\) \(2.007950949\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 \)
good3 \( 1 + (0.939 + 0.342i)T^{2} \)
5 \( 1 + (-0.473 - 0.397i)T + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.280 - 1.59i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (-1.52 + 0.553i)T + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (1.52 + 0.553i)T + (0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 0.618T + T^{2} \)
41 \( 1 + (0.107 - 0.608i)T + (-0.939 - 0.342i)T^{2} \)
43 \( 1 + (-0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 + (0.473 - 0.397i)T + (0.173 - 0.984i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (1.23 - 1.04i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (-0.107 + 0.608i)T + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.107 + 0.608i)T + (-0.939 + 0.342i)T^{2} \)
97 \( 1 + (1.52 - 0.553i)T + (0.766 - 0.642i)T^{2} \)
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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

−9.671332521210169977892011127355, −9.174601749312054818087718780487, −7.88436225515720016165497696312, −6.96312926370642743081648677156, −6.13939079854093484665350719257, −5.63056150096935075320905505067, −4.54347760472291183901769002211, −3.55805458109715855691361100858, −2.71126620787086906439744874980, −1.61205313937249591726604730771, 1.72375609122142383941560053139, 3.07600560339644968896983209855, 3.64117185564408647250120226607, 5.28570000664308299139072990897, 5.38399411451533275030919395810, 6.17128619299666342245971746841, 7.43182598798615811657779144185, 8.025976441242058752266083692730, 8.742523088544711655132981286625, 9.878791088042973451981796150152

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