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Properties

Label 2-19e2-19.12-c2-0-43
Degree $2$
Conductor $361$
Sign $-0.977 - 0.211i$
Analytic cond. $9.83653$
Root an. cond. $3.13632$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.12 − 1.80i)2-s + (−3.12 + 1.80i)3-s + (4.5 − 7.79i)4-s + (−2 − 3.46i)5-s + (−6.5 + 11.2i)6-s − 5·7-s − 18.0i·8-s + (2 − 3.46i)9-s + (−12.4 − 7.21i)10-s − 10·11-s + 32.4i·12-s + (−3.12 − 1.80i)13-s + (−15.6 + 9.01i)14-s + (12.4 + 7.21i)15-s + (−14.5 − 25.1i)16-s + (−7.5 − 12.9i)17-s + ⋯
L(s)  = 1  + (1.56 − 0.901i)2-s + (−1.04 + 0.600i)3-s + (1.12 − 1.94i)4-s + (−0.400 − 0.692i)5-s + (−1.08 + 1.87i)6-s − 0.714·7-s − 2.25i·8-s + (0.222 − 0.384i)9-s + (−1.24 − 0.721i)10-s − 0.909·11-s + 2.70i·12-s + (−0.240 − 0.138i)13-s + (−1.11 + 0.643i)14-s + (0.832 + 0.480i)15-s + (−0.906 − 1.56i)16-s + (−0.441 − 0.764i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.977 - 0.211i$
Analytic conductor: \(9.83653\)
Root analytic conductor: \(3.13632\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 361,\ (\ :1),\ -0.977 - 0.211i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.123697 + 1.15788i\)
\(L(\frac12)\) \(\approx\) \(0.123697 + 1.15788i\)
\(L(2)\) 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 \( 1 + (-3.12 + 1.80i)T + (2 - 3.46i)T^{2} \)
3 \( 1 + (3.12 - 1.80i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (2 + 3.46i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 5T + 49T^{2} \)
11 \( 1 + 10T + 121T^{2} \)
13 \( 1 + (3.12 + 1.80i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (7.5 + 12.9i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (17.5 - 30.3i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (15.6 + 9.01i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 36.0iT - 961T^{2} \)
37 \( 1 + 21.6iT - 1.36e3T^{2} \)
41 \( 1 + (-31.2 + 18.0i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-10 - 17.3i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (5 - 8.66i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-65.5 - 37.8i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-15.6 + 9.01i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-20 + 34.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (34.3 + 19.8i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-93.6 + 54.0i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (52.5 + 90.9i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (31.2 - 18.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 40T + 6.88e3T^{2} \)
89 \( 1 + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-106. + 61.2i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−11.09535319185800600210804233982, −10.23188761458071928004240294032, −9.440440936235322994209310671286, −7.65869932553058423402139034523, −6.15257501643265000831503872658, −5.45231696645507253538154547248, −4.69011014358483051653908703082, −3.82319937266820744334255447300, −2.45567404469774321006084564884, −0.33168053368596443770838699718, 2.70736317452711992804966813205, 3.86188322478277541559634573788, 5.10293803599278816223949499799, 5.99081667550147139629672663850, 6.72267184123307630406626914669, 7.24401821741758027850053508175, 8.413221128415065629149549242774, 10.31674796674678346967221580457, 11.19548604228426035268045616689, 12.05096981286000418511103104637

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