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Properties

Label 2-19e2-19.11-c1-0-19
Degree $2$
Conductor $361$
Sign $-0.928 + 0.370i$
Analytic cond. $2.88259$
Root an. cond. $1.69782$
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  + (1.26 − 2.19i)2-s + (0.326 − 0.565i)3-s + (−2.20 − 3.82i)4-s + (0.673 − 1.16i)5-s + (−0.826 − 1.43i)6-s + 1.53·7-s − 6.10·8-s + (1.28 + 2.22i)9-s + (−1.70 − 2.95i)10-s − 1.18·11-s − 2.87·12-s + (−1.35 − 2.35i)13-s + (1.93 − 3.35i)14-s + (−0.439 − 0.761i)15-s + (−3.31 + 5.74i)16-s + (−1.93 + 3.35i)17-s + ⋯
L(s)  = 1  + (0.895 − 1.55i)2-s + (0.188 − 0.326i)3-s + (−1.10 − 1.91i)4-s + (0.301 − 0.521i)5-s + (−0.337 − 0.584i)6-s + 0.579·7-s − 2.15·8-s + (0.428 + 0.743i)9-s + (−0.539 − 0.934i)10-s − 0.357·11-s − 0.831·12-s + (−0.376 − 0.652i)13-s + (0.518 − 0.897i)14-s + (−0.113 − 0.196i)15-s + (−0.829 + 1.43i)16-s + (−0.470 + 0.814i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 + 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.928 + 0.370i$
Analytic conductor: \(2.88259\)
Root analytic conductor: \(1.69782\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 361,\ (\ :1/2),\ -0.928 + 0.370i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.423769 - 2.20331i\)
\(L(\frac12)\) \(\approx\) \(0.423769 - 2.20331i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (-1.26 + 2.19i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.326 + 0.565i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.673 + 1.16i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 1.53T + 7T^{2} \)
11 \( 1 + 1.18T + 11T^{2} \)
13 \( 1 + (1.35 + 2.35i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.93 - 3.35i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.53 - 4.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.32 - 4.02i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.83T + 31T^{2} \)
37 \( 1 + 4.10T + 37T^{2} \)
41 \( 1 + (-4.99 + 8.64i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.35 + 7.53i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.286 + 0.497i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.47 + 2.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.96 + 3.40i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.25 - 3.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.94 - 3.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.46 + 6.00i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.06 - 5.30i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.90 - 8.49i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.3T + 83T^{2} \)
89 \( 1 + (-1.21 - 2.10i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.68 - 6.38i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.91091265203617278194071374221, −10.62456553697714499837495281859, −9.493540108884729648750239996965, −8.475869754938646549734249192681, −7.26675397289342727389302086639, −5.44521634149830749700769607079, −5.00163610972480703962402769544, −3.74804974754259904503006531201, −2.36536658173243725557177759024, −1.40266350574039103710252165398, 2.84715859889982907217347872798, 4.29286014471803721553004857802, 4.89814154157425354133355924414, 6.24953451420290035858520066245, 6.85255097278429040158337275220, 7.78986979120373796083984265609, 8.819419678099156049393172809521, 9.720861304842630512533234396477, 11.00855916253117346121103876273, 12.16331682017778446599167341785

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