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Properties

Label 2-19e2-19.11-c1-0-4
Degree $2$
Conductor $361$
Sign $-0.980 - 0.194i$
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  + (−1 + 1.73i)3-s + (1 + 1.73i)4-s + (−1.5 + 2.59i)5-s − 7-s + (−0.499 − 0.866i)9-s + 3·11-s − 3.99·12-s + (−2 − 3.46i)13-s + (−3 − 5.19i)15-s + (−1.99 + 3.46i)16-s + (1.5 − 2.59i)17-s − 6·20-s + (1 − 1.73i)21-s + (−2 − 3.46i)25-s − 4.00·27-s + (−1 − 1.73i)28-s + ⋯
L(s)  = 1  + (−0.577 + 0.999i)3-s + (0.5 + 0.866i)4-s + (−0.670 + 1.16i)5-s − 0.377·7-s + (−0.166 − 0.288i)9-s + 0.904·11-s − 1.15·12-s + (−0.554 − 0.960i)13-s + (−0.774 − 1.34i)15-s + (−0.499 + 0.866i)16-s + (0.363 − 0.630i)17-s − 1.34·20-s + (0.218 − 0.377i)21-s + (−0.400 − 0.692i)25-s − 0.769·27-s + (−0.188 − 0.327i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.980 - 0.194i$
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: \((2,\ 361,\ (\ :1/2),\ -0.980 - 0.194i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0925384 + 0.942286i\)
\(L(\frac12)\) \(\approx\) \(0.0925384 + 0.942286i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-6 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4 + 6.92i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.78564643265642416191800822427, −10.87650387715444720767429639264, −10.35829499850410731958478279308, −9.281765300152751264243825706669, −7.917478800658657964275367468724, −7.14133870709948718810079847702, −6.24708098653908182802901375709, −4.78218804942028138109042237150, −3.61470605393333156396552720171, −2.90797900916271043042589843193, 0.70563409124300056520875835495, 1.84784319427212481092452803250, 4.06546104084453705822207894063, 5.23792111080427081960031661389, 6.37543074311321838297649319913, 6.89878281894650312149549038205, 8.088913462501872611033346201758, 9.202706565184491915208549412588, 10.04706107116885528321319016114, 11.44355792141575607672674852206

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