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Properties

Label 2-2736-19.7-c1-0-19
Degree $2$
Conductor $2736$
Sign $0.0977 - 0.995i$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
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)5-s + 3·7-s + (−2.5 + 4.33i)13-s + (−2 − 3.46i)17-s + (4 + 1.73i)19-s + (−3 + 5.19i)23-s + (0.500 − 0.866i)25-s + (−2 + 3.46i)29-s + 7·31-s + (3 + 5.19i)35-s − 37-s + (5.5 + 9.52i)43-s + (−3 + 5.19i)47-s + 2·49-s + (−1 + 1.73i)53-s + ⋯
L(s)  = 1  + (0.447 + 0.774i)5-s + 1.13·7-s + (−0.693 + 1.20i)13-s + (−0.485 − 0.840i)17-s + (0.917 + 0.397i)19-s + (−0.625 + 1.08i)23-s + (0.100 − 0.173i)25-s + (−0.371 + 0.643i)29-s + 1.25·31-s + (0.507 + 0.878i)35-s − 0.164·37-s + (0.838 + 1.45i)43-s + (−0.437 + 0.757i)47-s + 0.285·49-s + (−0.137 + 0.237i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.0977 - 0.995i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.0977 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.049192051\)
\(L(\frac12)\) \(\approx\) \(2.049192051\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + (-4 - 1.73i)T \)
good5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.5 - 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.5 + 7.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16T + 83T^{2} \)
89 \( 1 + (-6 + 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7 - 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.281968304190846881382656072117, −8.034341938820332371636715427319, −7.54427566591219358721419870725, −6.74792265808836219270128844740, −6.01513035474501692563058594377, −4.97222143434921049825181007593, −4.47351210385094920700078117889, −3.21976151845715936982518385080, −2.29204719660334904828069524856, −1.42155051377593773348860172898, 0.67347671895627927502237874484, 1.79780884082885014557631323343, 2.72277130391418595595419827095, 4.03989490455344729230709226881, 4.88968745443964316322907829406, 5.36491778444830718988827758542, 6.20876152325514070858790975166, 7.29647068910001796852401970758, 8.056498021957868892901208731156, 8.511535382787352037929656385904

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