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Properties

Label 2-19e2-19.3-c2-0-11
Degree $2$
Conductor $361$
Sign $-0.775 - 0.631i$
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  + (0.694 + 3.93i)4-s + (−1.56 + 8.86i)5-s + (2.5 + 4.33i)7-s + (6.89 − 5.78i)9-s + (−1.5 + 2.59i)11-s + (−15.0 + 5.47i)16-s + (11.4 + 9.64i)17-s − 35.9·20-s + (−5.20 − 29.5i)23-s + (−52.6 − 19.1i)25-s + (−15.3 + 12.8i)28-s + (−42.2 + 15.3i)35-s + (27.5 + 23.1i)36-s + (−14.7 + 83.7i)43-s + (−11.2 − 4.10i)44-s + (40.5 + 70.1i)45-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)4-s + (−0.312 + 1.77i)5-s + (0.357 + 0.618i)7-s + (0.766 − 0.642i)9-s + (−0.136 + 0.236i)11-s + (−0.939 + 0.342i)16-s + (0.675 + 0.567i)17-s − 1.79·20-s + (−0.226 − 1.28i)23-s + (−2.10 − 0.766i)25-s + (−0.547 + 0.459i)28-s + (−1.20 + 0.439i)35-s + (0.766 + 0.642i)36-s + (−0.343 + 1.94i)43-s + (−0.256 − 0.0932i)44-s + (0.900 + 1.55i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.775 - 0.631i$
Analytic conductor: \(9.83653\)
Root analytic conductor: \(3.13632\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 361,\ (\ :1),\ -0.775 - 0.631i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.538552 + 1.51528i\)
\(L(\frac12)\) \(\approx\) \(0.538552 + 1.51528i\)
\(L(2)\) 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 \( 1 + (-0.694 - 3.93i)T^{2} \)
3 \( 1 + (-6.89 + 5.78i)T^{2} \)
5 \( 1 + (1.56 - 8.86i)T + (-23.4 - 8.55i)T^{2} \)
7 \( 1 + (-2.5 - 4.33i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-129. - 108. i)T^{2} \)
17 \( 1 + (-11.4 - 9.64i)T + (50.1 + 284. i)T^{2} \)
23 \( 1 + (5.20 + 29.5i)T + (-497. + 180. i)T^{2} \)
29 \( 1 + (-146. + 828. i)T^{2} \)
31 \( 1 + (480.5 - 832. i)T^{2} \)
37 \( 1 - 1.36e3T^{2} \)
41 \( 1 + (-1.28e3 + 1.08e3i)T^{2} \)
43 \( 1 + (14.7 - 83.7i)T + (-1.73e3 - 632. i)T^{2} \)
47 \( 1 + (-57.4 + 48.2i)T + (383. - 2.17e3i)T^{2} \)
53 \( 1 + (2.63e3 - 960. i)T^{2} \)
59 \( 1 + (-604. - 3.42e3i)T^{2} \)
61 \( 1 + (-17.8 - 101. i)T + (-3.49e3 + 1.27e3i)T^{2} \)
67 \( 1 + (-779. + 4.42e3i)T^{2} \)
71 \( 1 + (4.73e3 + 1.72e3i)T^{2} \)
73 \( 1 + (-23.4 + 8.55i)T + (4.08e3 - 3.42e3i)T^{2} \)
79 \( 1 + (-4.78e3 + 4.01e3i)T^{2} \)
83 \( 1 + (45 + 77.9i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-6.06e3 - 5.09e3i)T^{2} \)
97 \( 1 + (-1.63e3 - 9.26e3i)T^{2} \)
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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

−11.68791315565276588803176938041, −10.70304920878814698148569529300, −9.951662986444091024877986453207, −8.610410260757412525173217703671, −7.66358574129214106562217491340, −6.94264931928856109725913890944, −6.12128170654904841551797503823, −4.26488197866386400200984401862, −3.30010108396515225052034807420, −2.31196709525547666550636847277, 0.76379103569050361152957760738, 1.70832984929521151467411705682, 4.06742171781184565653492882397, 5.00014950328319283403628301696, 5.59751753308644209111878921205, 7.23871935495065178201207050795, 8.008418634122973410050616547218, 9.148040839546306460362209086886, 9.873506769407068856138965919926, 10.82016764038011479237852755088

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