Properties

Label 2-19e2-19.7-c1-0-20
Degree $2$
Conductor $361$
Sign $-0.143 - 0.989i$
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  + (−0.673 − 1.16i)2-s + (−1.43 − 2.49i)3-s + (0.0923 − 0.160i)4-s + (−0.439 − 0.761i)5-s + (−1.93 + 3.35i)6-s + 0.347·7-s − 2.94·8-s + (−2.64 + 4.58i)9-s + (−0.592 + 1.02i)10-s − 2.22·11-s − 0.532·12-s + (1.28 − 2.22i)13-s + (−0.233 − 0.405i)14-s + (−1.26 + 2.19i)15-s + (1.79 + 3.11i)16-s + (−0.233 − 0.405i)17-s + ⋯
L(s)  = 1  + (−0.476 − 0.825i)2-s + (−0.831 − 1.43i)3-s + (0.0461 − 0.0800i)4-s + (−0.196 − 0.340i)5-s + (−0.791 + 1.37i)6-s + 0.131·7-s − 1.04·8-s + (−0.881 + 1.52i)9-s + (−0.187 + 0.324i)10-s − 0.671·11-s − 0.153·12-s + (0.356 − 0.618i)13-s + (−0.0625 − 0.108i)14-s + (−0.326 + 0.566i)15-s + (0.449 + 0.778i)16-s + (−0.0567 − 0.0982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.143 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.143 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.143 - 0.989i$
Analytic conductor: \(2.88259\)
Root analytic conductor: \(1.69782\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (292, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 361,\ (\ :1/2),\ -0.143 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.306729 + 0.354262i\)
\(L(\frac12)\) \(\approx\) \(0.306729 + 0.354262i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.673 + 1.16i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.43 + 2.49i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.439 + 0.761i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 0.347T + 7T^{2} \)
11 \( 1 + 2.22T + 11T^{2} \)
13 \( 1 + (-1.28 + 2.22i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.233 + 0.405i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.34 + 2.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.43 - 5.95i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.10T + 31T^{2} \)
37 \( 1 + 4.94T + 37T^{2} \)
41 \( 1 + (-1.23 - 2.14i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.95 + 3.37i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.64 + 6.31i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.41 - 2.45i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.15 + 5.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.56 - 7.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.83 + 6.64i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.65 + 8.05i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.694 + 1.20i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.92 + 10.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.8T + 83T^{2} \)
89 \( 1 + (5.14 - 8.91i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.72 - 8.18i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.86838826716310060184125841461, −10.27371693383038587872794316873, −8.819209715309176695028527782196, −8.007887452813317127457305148770, −6.90059784541803512725482624681, −6.00318846831061244120932032466, −5.04781520362783428027811079596, −2.90016522155440990288185943735, −1.60710616291669339673987143838, −0.41154943998556114603173191184, 3.11603123310430425953184073363, 4.29351881543810221085817425288, 5.44076485295806675484023674404, 6.29509012405236765834163679057, 7.36063445743197331763510029527, 8.449565301826259393192732316806, 9.382635847244580693895720136497, 10.15521877974576313385977435140, 11.24139494308391081111561730698, 11.57717499521715817987462993750

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