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Properties

Label 2-19e2-19.11-c1-0-15
Degree $2$
Conductor $361$
Sign $0.599 + 0.800i$
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  + (0.439 − 0.761i)2-s + (0.266 − 0.460i)3-s + (0.613 + 1.06i)4-s + (1.26 − 2.19i)5-s + (−0.233 − 0.405i)6-s − 1.87·7-s + 2.83·8-s + (1.35 + 2.35i)9-s + (−1.11 − 1.92i)10-s + 3.41·11-s + 0.652·12-s + (−2.64 − 4.58i)13-s + (−0.826 + 1.43i)14-s + (−0.673 − 1.16i)15-s + (0.0209 − 0.0362i)16-s + (−0.826 + 1.43i)17-s + ⋯
L(s)  = 1  + (0.310 − 0.538i)2-s + (0.153 − 0.266i)3-s + (0.306 + 0.531i)4-s + (0.566 − 0.980i)5-s + (−0.0955 − 0.165i)6-s − 0.710·7-s + 1.00·8-s + (0.452 + 0.784i)9-s + (−0.352 − 0.609i)10-s + 1.02·11-s + 0.188·12-s + (−0.733 − 1.27i)13-s + (−0.220 + 0.382i)14-s + (−0.173 − 0.301i)15-s + (0.00523 − 0.00906i)16-s + (−0.200 + 0.347i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $0.599 + 0.800i$
Analytic conductor: \(2.88259\)
Root analytic conductor: \(1.69782\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 361,\ (\ :1/2),\ 0.599 + 0.800i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75002 - 0.875173i\)
\(L(\frac12)\) \(\approx\) \(1.75002 - 0.875173i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.439 + 0.761i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.266 + 0.460i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.26 + 2.19i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.87T + 7T^{2} \)
11 \( 1 - 3.41T + 11T^{2} \)
13 \( 1 + (2.64 + 4.58i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.826 - 1.43i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.879 + 1.52i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.73 + 3.00i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.94T + 31T^{2} \)
37 \( 1 - 0.837T + 37T^{2} \)
41 \( 1 + (2.24 - 3.88i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.40 - 4.15i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.358 + 0.620i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.05 + 5.28i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.37 - 9.31i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.19 + 3.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.10 - 12.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.87 - 11.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.75 + 6.51i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.48 - 6.03i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.51T + 83T^{2} \)
89 \( 1 + (1.14 + 1.97i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.911 - 1.57i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.49221246562205196095968768776, −10.35976520639774844439441589488, −9.620186581213495931985376863056, −8.464635804730811777817020569701, −7.61463171219245378254053459283, −6.53251262276352532109439366334, −5.22087292663419402273064393518, −4.15128647879574296946410012581, −2.82068740628152992517793540383, −1.54976932439311631903015362400, 1.90208274629775313726054877896, 3.45963469748350464469398619784, 4.66774918571958173674546271606, 6.13676541557135144327713029794, 6.68576858175457192870233010833, 7.22755332267483587283083083655, 9.219045213950491379931276149631, 9.634675482151707966920619238319, 10.51395562556788941326233440201, 11.50326372783637264756619615880

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