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Properties

Label 2-1254-19.11-c1-0-10
Degree $2$
Conductor $1254$
Sign $0.813 - 0.582i$
Analytic cond. $10.0132$
Root an. cond. $3.16437$
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  + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s − 7-s − 0.999·8-s + (−0.499 − 0.866i)9-s − 11-s + 0.999·12-s + (2 + 3.46i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (1.5 − 2.59i)17-s − 0.999·18-s + (−0.5 + 4.33i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.204 + 0.353i)6-s − 0.377·7-s − 0.353·8-s + (−0.166 − 0.288i)9-s − 0.301·11-s + 0.288·12-s + (0.554 + 0.960i)13-s + (−0.133 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (0.363 − 0.630i)17-s − 0.235·18-s + (−0.114 + 0.993i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1254\)    =    \(2 \cdot 3 \cdot 11 \cdot 19\)
Sign: $0.813 - 0.582i$
Analytic conductor: \(10.0132\)
Root analytic conductor: \(3.16437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1254} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1254,\ (\ :1/2),\ 0.813 - 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.439306946\)
\(L(\frac12)\) \(\approx\) \(1.439306946\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + T \)
19 \( 1 + (0.5 - 4.33i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + T + 7T^{2} \)
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 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.5 + 6.06i)T + (-21.5 - 37.2i)T^{2} \)
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 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.5 + 7.79i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.5 - 9.52i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.937615253785856090071833122571, −9.149266971101737756660440138579, −8.425887385706156858017927079795, −7.14704104904295897991657053997, −6.28269199112625360851071597890, −5.41863377542927728609171288729, −4.53736144114000981193898448338, −3.66740614331736106506027672789, −2.74253233876165412333322618281, −1.27075126801111658495873671764, 0.62644032408619753879932427441, 2.44551273958970450907355079014, 3.50799744926839791682923743595, 4.64360159764388942719442269385, 5.59900790549217667417471231787, 6.28025574424133776786204982113, 6.97951230279840894964243277712, 8.077084450076726766986258696292, 8.349909152167721560154844690951, 9.620001934421596755830855360690

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