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Properties

Label 2-912-228.131-c1-0-22
Degree $2$
Conductor $912$
Sign $0.270 + 0.962i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
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  + (−1.11 + 1.32i)3-s + (−4.52 + 2.61i)7-s + (−0.520 − 2.95i)9-s + (−0.375 + 0.315i)13-s + (−0.5 − 4.33i)19-s + (1.57 − 8.91i)21-s + (3.83 − 3.21i)25-s + (4.5 + 2.59i)27-s + (4.97 − 2.87i)31-s − 8.55·37-s − 0.849i·39-s + (−2.07 − 5.70i)43-s + (10.1 − 17.5i)49-s + (6.30 + 4.15i)57-s + (10.1 + 3.68i)61-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)3-s + (−1.71 + 0.987i)7-s + (−0.173 − 0.984i)9-s + (−0.104 + 0.0873i)13-s + (−0.114 − 0.993i)19-s + (0.342 − 1.94i)21-s + (0.766 − 0.642i)25-s + (0.866 + 0.499i)27-s + (0.893 − 0.515i)31-s − 1.40·37-s − 0.135i·39-s + (−0.316 − 0.870i)43-s + (1.44 − 2.51i)49-s + (0.834 + 0.550i)57-s + (1.29 + 0.472i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.270 + 0.962i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (815, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.270 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.321667 - 0.243799i\)
\(L(\frac12)\) \(\approx\) \(0.321667 - 0.243799i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.11 - 1.32i)T \)
19 \( 1 + (0.5 + 4.33i)T \)
good5 \( 1 + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (4.52 - 2.61i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.375 - 0.315i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (-4.97 + 2.87i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.55T + 37T^{2} \)
41 \( 1 + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (2.07 + 5.70i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-10.1 - 3.68i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (12.7 - 2.24i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (10.8 + 9.08i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-10.1 + 12.1i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-15.4 + 87.6i)T^{2} \)
97 \( 1 + (-3.29 + 18.7i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.01326578460084364738754638380, −9.104251466602363474976006179253, −8.709383937993828353906496178896, −7.00615448077623660261246963062, −6.38200503819866089590031739191, −5.61875021339141192305018967397, −4.65889977553599189344570510687, −3.47886478186583444576426132453, −2.63260837751531576867232137554, −0.22880649303927208142509817992, 1.19408425453849615831208732409, 2.86848151282677363848587909329, 3.87233968586282071726460677139, 5.16335961627371743149034889173, 6.21802444053199722454988718849, 6.77718802703471159986751616058, 7.45942746190544957643662023823, 8.484136263076838720886690157943, 9.659542240457837389413003694498, 10.31661846961244570908071752356

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