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Properties

Label 2-19e2-19.15-c2-0-4
Degree $2$
Conductor $361$
Sign $-0.990 + 0.135i$
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  + (3.06 + 2.57i)4-s + (−6.89 + 5.78i)5-s + (2.5 − 4.33i)7-s + (−8.45 + 3.07i)9-s + (−1.5 − 2.59i)11-s + (2.77 + 15.7i)16-s + (−14.0 − 5.13i)17-s − 36·20-s + (−22.9 − 19.2i)23-s + (9.72 − 55.1i)25-s + (18.7 − 6.84i)28-s + (7.81 + 44.3i)35-s + (−33.8 − 12.3i)36-s + (−65.1 + 54.6i)43-s + (2.08 − 11.8i)44-s + (40.5 − 70.1i)45-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)4-s + (−1.37 + 1.15i)5-s + (0.357 − 0.618i)7-s + (−0.939 + 0.342i)9-s + (−0.136 − 0.236i)11-s + (0.173 + 0.984i)16-s + (−0.829 − 0.301i)17-s − 1.80·20-s + (−0.999 − 0.838i)23-s + (0.388 − 2.20i)25-s + (0.671 − 0.244i)28-s + (0.223 + 1.26i)35-s + (−0.939 − 0.342i)36-s + (−1.51 + 1.27i)43-s + (0.0473 − 0.268i)44-s + (0.900 − 1.55i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.990 + 0.135i$
Analytic conductor: \(9.83653\)
Root analytic conductor: \(3.13632\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 361,\ (\ :1),\ -0.990 + 0.135i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0312592 - 0.460480i\)
\(L(\frac12)\) \(\approx\) \(0.0312592 - 0.460480i\)
\(L(2)\) 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 \( 1 + (-3.06 - 2.57i)T^{2} \)
3 \( 1 + (8.45 - 3.07i)T^{2} \)
5 \( 1 + (6.89 - 5.78i)T + (4.34 - 24.6i)T^{2} \)
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 \( 1 + (158. + 57.8i)T^{2} \)
17 \( 1 + (14.0 + 5.13i)T + (221. + 185. i)T^{2} \)
23 \( 1 + (22.9 + 19.2i)T + (91.8 + 520. i)T^{2} \)
29 \( 1 + (-644. + 540. i)T^{2} \)
31 \( 1 + (480.5 + 832. i)T^{2} \)
37 \( 1 - 1.36e3T^{2} \)
41 \( 1 + (1.57e3 - 574. i)T^{2} \)
43 \( 1 + (65.1 - 54.6i)T + (321. - 1.82e3i)T^{2} \)
47 \( 1 + (70.4 - 25.6i)T + (1.69e3 - 1.41e3i)T^{2} \)
53 \( 1 + (-487. - 2.76e3i)T^{2} \)
59 \( 1 + (-2.66e3 - 2.23e3i)T^{2} \)
61 \( 1 + (-78.9 - 66.2i)T + (646. + 3.66e3i)T^{2} \)
67 \( 1 + (-3.43e3 + 2.88e3i)T^{2} \)
71 \( 1 + (-875. + 4.96e3i)T^{2} \)
73 \( 1 + (4.34 + 24.6i)T + (-5.00e3 + 1.82e3i)T^{2} \)
79 \( 1 + (5.86e3 - 2.13e3i)T^{2} \)
83 \( 1 + (45 - 77.9i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (7.44e3 + 2.70e3i)T^{2} \)
97 \( 1 + (-7.20e3 - 6.04e3i)T^{2} \)
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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

−11.32671678080138862842022919185, −11.20758213618700498190137643907, −10.27466320229688226894440468986, −8.391748463514147996982802888822, −7.966040912032460859881776116226, −7.06022219776041298085006478629, −6.29461268180005976669596978958, −4.45793368459580694717336968715, −3.41086117369192241314503784563, −2.51019718717878952234215219978, 0.19196118235138939107039681560, 1.89626795626744840952552928430, 3.54288324437226850556931275494, 4.89195366162951842678476684008, 5.69415758583714943348640467545, 6.94764451553472551802695312804, 8.135354392138121175378809522055, 8.650327798292444791631869989638, 9.763895922944134653569353938276, 11.10546379592715395420193572978

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