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Properties

Label 2-38e2-19.11-c1-0-24
Degree $2$
Conductor $1444$
Sign $-0.658 + 0.752i$
Analytic cond. $11.5303$
Root an. cond. $3.39564$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)3-s + (0.5 − 0.866i)5-s − 3·7-s + (−0.499 − 0.866i)9-s + 5·11-s + (−2 − 3.46i)13-s + (−0.999 − 1.73i)15-s + (1.5 − 2.59i)17-s + (−3 + 5.19i)21-s + (−4 − 6.92i)23-s + (2 + 3.46i)25-s + 4.00·27-s + (−1 − 1.73i)29-s − 4·31-s + (5 − 8.66i)33-s + ⋯
L(s)  = 1  + (0.577 − 0.999i)3-s + (0.223 − 0.387i)5-s − 1.13·7-s + (−0.166 − 0.288i)9-s + 1.50·11-s + (−0.554 − 0.960i)13-s + (−0.258 − 0.447i)15-s + (0.363 − 0.630i)17-s + (−0.654 + 1.13i)21-s + (−0.834 − 1.44i)23-s + (0.400 + 0.692i)25-s + 0.769·27-s + (−0.185 − 0.321i)29-s − 0.718·31-s + (0.870 − 1.50i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $-0.658 + 0.752i$
Analytic conductor: \(11.5303\)
Root analytic conductor: \(3.39564\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :1/2),\ -0.658 + 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.728678575\)
\(L(\frac12)\) \(\approx\) \(1.728678575\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 3T + 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + (-5 + 8.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1 + 1.73i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.5 - 7.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-6 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4 - 6.92i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.109934664024440756609710152654, −8.520097275848638692511335080616, −7.47712127597511159708299247554, −6.90598270335006147575555173601, −6.17058295471821885490556011945, −5.16784165854402464292140616481, −3.87460224512887069871937373280, −2.95591131984208166590982447316, −1.91142896603528396137157560071, −0.64101431471754566848328504687, 1.72357619854808491325799253027, 3.14753127930463757655219530761, 3.73639881935950836076735897208, 4.45389087770666232967265242029, 5.81077736990966405904312623579, 6.59083925171028664314499355053, 7.22228649559701386862291507096, 8.578078489088206695794602771094, 9.231347012229685998648068009846, 9.760579063697403803129108711091

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