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Properties

Label 2-1338-223.39-c1-0-27
Degree $2$
Conductor $1338$
Sign $0.942 + 0.333i$
Analytic cond. $10.6839$
Root an. cond. $3.26863$
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  + 2-s + (0.5 − 0.866i)3-s + 4-s + (0.869 + 1.50i)5-s + (0.5 − 0.866i)6-s + 2.51·7-s + 8-s + (−0.499 − 0.866i)9-s + (0.869 + 1.50i)10-s + (−1.86 − 3.22i)11-s + (0.5 − 0.866i)12-s + 1.79·13-s + 2.51·14-s + 1.73·15-s + 16-s + 3.12·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.288 − 0.499i)3-s + 0.5·4-s + (0.388 + 0.673i)5-s + (0.204 − 0.353i)6-s + 0.951·7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.274 + 0.476i)10-s + (−0.561 − 0.972i)11-s + (0.144 − 0.249i)12-s + 0.498·13-s + 0.672·14-s + 0.449·15-s + 0.250·16-s + 0.757·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1338\)    =    \(2 \cdot 3 \cdot 223\)
Sign: $0.942 + 0.333i$
Analytic conductor: \(10.6839\)
Root analytic conductor: \(3.26863\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1338} (931, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1338,\ (\ :1/2),\ 0.942 + 0.333i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.403810966\)
\(L(\frac12)\) \(\approx\) \(3.403810966\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.5 + 0.866i)T \)
223 \( 1 + (10.1 - 10.9i)T \)
good5 \( 1 + (-0.869 - 1.50i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 2.51T + 7T^{2} \)
11 \( 1 + (1.86 + 3.22i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.79T + 13T^{2} \)
17 \( 1 - 3.12T + 17T^{2} \)
19 \( 1 + (1.42 - 2.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.670 + 1.16i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.86 + 4.95i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.73 - 6.47i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.06 - 5.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.4T + 41T^{2} \)
43 \( 1 + (-0.756 + 1.31i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.05 - 3.56i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.89 + 8.48i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 0.483T + 59T^{2} \)
61 \( 1 + (5.76 - 9.98i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.93 - 10.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.98 + 3.43i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.82 + 13.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.57 - 7.91i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.88 - 15.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.09 - 1.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.59 + 16.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.644989309260686426450196405149, −8.426932945297180929856421985602, −7.965120543547768780646378042095, −7.07180709637808988362916209748, −6.07291919819390332498568785579, −5.61400113782015816447804995504, −4.42163469232078419380251790541, −3.30802298360534521490940348722, −2.50207978410468364569162559326, −1.32958589284247958509484768889, 1.48763429030698527076617161412, 2.50153093130540627985094475878, 3.79088193302759801995227905949, 4.67989751580099044070900741499, 5.22364463238727702674785201327, 6.02243184778505310022564221718, 7.44252992773801982060722633553, 7.85006745315788536875920493039, 9.053027757797346594234818394493, 9.487493117290581952196760847774

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