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Properties

Label 2-38e2-19.12-c0-0-2
Degree $2$
Conductor $1444$
Sign $0.305 + 0.952i$
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  + (1.22 − 0.707i)3-s + (−0.5 − 0.866i)5-s + 7-s + (0.499 − 0.866i)9-s − 11-s + (−1.22 − 0.707i)15-s + (−0.5 − 0.866i)17-s + (1.22 − 0.707i)21-s + (1.22 + 0.707i)29-s + (−1.22 + 0.707i)33-s + (−0.5 − 0.866i)35-s + 1.41i·37-s + (−1.22 + 0.707i)41-s + (0.5 + 0.866i)43-s − 0.999·45-s + ⋯
L(s)  = 1  + (1.22 − 0.707i)3-s + (−0.5 − 0.866i)5-s + 7-s + (0.499 − 0.866i)9-s − 11-s + (−1.22 − 0.707i)15-s + (−0.5 − 0.866i)17-s + (1.22 − 0.707i)21-s + (1.22 + 0.707i)29-s + (−1.22 + 0.707i)33-s + (−0.5 − 0.866i)35-s + 1.41i·37-s + (−1.22 + 0.707i)41-s + (0.5 + 0.866i)43-s − 0.999·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $0.305 + 0.952i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ 0.305 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.551044789\)
\(L(\frac12)\) \(\approx\) \(1.551044789\)
\(L(1)\) 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 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.41iT - T^{2} \)
41 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.213027246506958113225833932738, −8.463975772410365110799749816516, −8.130377619555141864098468056684, −7.49691225469888413304683872033, −6.55602780678835277494284934907, −4.99317592202028149222310898801, −4.71433398809704680065282270064, −3.26266263314914305793542276954, −2.38042933383133406369106114126, −1.25124905742411499954093168927, 2.09910802855567735881715533817, 2.89695399801344592744918506255, 3.84548206416847886229242945046, 4.55787058050196450633522663542, 5.63426913553017837733276834306, 6.89681608510278422906103186813, 7.75814896212571315039642453574, 8.269443277517704201362133246899, 8.908175404471205780294998547981, 9.938918849197730665610657612435

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