Properties

Label 2-19e2-1.1-c1-0-12
Degree $2$
Conductor $361$
Sign $1$
Analytic cond. $2.88259$
Root an. cond. $1.69782$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.34·2-s + 2.87·3-s − 0.184·4-s + 0.879·5-s + 3.87·6-s + 0.347·7-s − 2.94·8-s + 5.29·9-s + 1.18·10-s − 2.22·11-s − 0.532·12-s − 2.57·13-s + 0.467·14-s + 2.53·15-s − 3.59·16-s + 0.467·17-s + 7.12·18-s − 0.162·20-s + 21-s − 3·22-s − 2.69·23-s − 8.47·24-s − 4.22·25-s − 3.46·26-s + 6.59·27-s − 0.0641·28-s + 6.87·29-s + ⋯
L(s)  = 1  + 0.952·2-s + 1.66·3-s − 0.0923·4-s + 0.393·5-s + 1.58·6-s + 0.131·7-s − 1.04·8-s + 1.76·9-s + 0.374·10-s − 0.671·11-s − 0.153·12-s − 0.713·13-s + 0.125·14-s + 0.653·15-s − 0.899·16-s + 0.113·17-s + 1.68·18-s − 0.0363·20-s + 0.218·21-s − 0.639·22-s − 0.561·23-s − 1.73·24-s − 0.845·25-s − 0.680·26-s + 1.26·27-s − 0.0121·28-s + 1.27·29-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(3.075532446\)
\(L(\frac12)\) \(\approx\) \(3.075532446\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.34T + 2T^{2} \)
3 \( 1 - 2.87T + 3T^{2} \)
5 \( 1 - 0.879T + 5T^{2} \)
7 \( 1 - 0.347T + 7T^{2} \)
11 \( 1 + 2.22T + 11T^{2} \)
13 \( 1 + 2.57T + 13T^{2} \)
17 \( 1 - 0.467T + 17T^{2} \)
23 \( 1 + 2.69T + 23T^{2} \)
29 \( 1 - 6.87T + 29T^{2} \)
31 \( 1 - 7.10T + 31T^{2} \)
37 \( 1 + 4.94T + 37T^{2} \)
41 \( 1 + 2.47T + 41T^{2} \)
43 \( 1 - 3.90T + 43T^{2} \)
47 \( 1 + 7.29T + 47T^{2} \)
53 \( 1 - 2.83T + 53T^{2} \)
59 \( 1 - 6.30T + 59T^{2} \)
61 \( 1 - 9.12T + 61T^{2} \)
67 \( 1 + 7.67T + 67T^{2} \)
71 \( 1 - 9.30T + 71T^{2} \)
73 \( 1 - 1.38T + 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 + 14.8T + 83T^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 + 9.45T + 97T^{2} \)
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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

−11.82847035227092728952560802422, −10.13895912933182563538853339236, −9.621167918096850934933523719004, −8.532441758409032066690774716175, −7.925179242709707989742441980582, −6.61698226712208514730901470996, −5.26288813311469045183077624962, −4.27223811406285223842955492809, −3.15907241361875228465613894042, −2.27155535338614360775181545455, 2.27155535338614360775181545455, 3.15907241361875228465613894042, 4.27223811406285223842955492809, 5.26288813311469045183077624962, 6.61698226712208514730901470996, 7.925179242709707989742441980582, 8.532441758409032066690774716175, 9.621167918096850934933523719004, 10.13895912933182563538853339236, 11.82847035227092728952560802422

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