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Properties

Label 2-2736-57.50-c1-0-33
Degree $2$
Conductor $2736$
Sign $0.414 + 0.910i$
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  + (2.99 − 1.72i)5-s − 0.128·7-s − 6.15i·11-s + (2.67 + 1.54i)13-s + (4.91 − 2.83i)17-s + (3.96 + 1.81i)19-s + (−2.18 − 1.26i)23-s + (3.46 − 5.99i)25-s + (−3.40 + 5.90i)29-s + 10.0i·31-s + (−0.383 + 0.221i)35-s − 10.1i·37-s + (2.34 + 4.05i)41-s + (−3.02 − 5.24i)43-s + (−0.266 − 0.153i)47-s + ⋯
L(s)  = 1  + (1.33 − 0.772i)5-s − 0.0484·7-s − 1.85i·11-s + (0.741 + 0.428i)13-s + (1.19 − 0.687i)17-s + (0.908 + 0.417i)19-s + (−0.456 − 0.263i)23-s + (0.692 − 1.19i)25-s + (−0.632 + 1.09i)29-s + 1.80i·31-s + (−0.0648 + 0.0374i)35-s − 1.67i·37-s + (0.365 + 0.633i)41-s + (−0.461 − 0.799i)43-s + (−0.0388 − 0.0224i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.414 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.414 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.414 + 0.910i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.414 + 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.553455845\)
\(L(\frac12)\) \(\approx\) \(2.553455845\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.96 - 1.81i)T \)
good5 \( 1 + (-2.99 + 1.72i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.128T + 7T^{2} \)
11 \( 1 + 6.15iT - 11T^{2} \)
13 \( 1 + (-2.67 - 1.54i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.91 + 2.83i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.18 + 1.26i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.40 - 5.90i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.0iT - 31T^{2} \)
37 \( 1 + 10.1iT - 37T^{2} \)
41 \( 1 + (-2.34 - 4.05i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.02 + 5.24i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.266 + 0.153i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.05 + 7.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.05 + 7.02i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.20 - 3.81i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.51 - 2.60i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.67 - 6.36i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.15 - 5.47i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.95 - 1.70i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.267iT - 83T^{2} \)
89 \( 1 + (-4.52 + 7.84i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (16.2 - 9.35i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.731925062033701731924470441765, −8.210353923793355644272288976107, −7.07433148988322193306117469923, −6.17443595727370927362233137829, −5.48224400042684021667046121080, −5.21648873513680149011312454249, −3.70296381170769972807720037613, −3.04762800701896204188492955856, −1.66679299149876645502292571103, −0.896980358300689569882728876355, 1.42497779413213706574352221448, 2.22361580964819889991543838936, 3.15728472494424473883953402338, 4.20840047125385545908028134343, 5.25302072574670760446657742453, 5.98661555436931907998410534474, 6.51089063871275068903650231500, 7.53603897701825436034461016372, 7.946991826703932832937104566134, 9.341423272449008196501336841275

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