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Properties

Label 2-2736-57.50-c1-0-17
Degree $2$
Conductor $2736$
Sign $0.261 - 0.965i$
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  + (−0.406 + 0.234i)5-s + 3.49·7-s + 1.34i·11-s + (4.30 + 2.48i)13-s + (−6.31 + 3.64i)17-s + (−1.88 − 3.92i)19-s + (6.97 + 4.02i)23-s + (−2.38 + 4.13i)25-s + (−4.74 + 8.21i)29-s − 3.95i·31-s + (−1.42 + 0.820i)35-s − 0.581i·37-s + (−0.761 − 1.31i)41-s + (4.63 + 8.02i)43-s + (1.06 + 0.615i)47-s + ⋯
L(s)  = 1  + (−0.181 + 0.105i)5-s + 1.32·7-s + 0.406i·11-s + (1.19 + 0.689i)13-s + (−1.53 + 0.884i)17-s + (−0.433 − 0.901i)19-s + (1.45 + 0.839i)23-s + (−0.477 + 0.827i)25-s + (−0.880 + 1.52i)29-s − 0.711i·31-s + (−0.240 + 0.138i)35-s − 0.0955i·37-s + (−0.118 − 0.205i)41-s + (0.706 + 1.22i)43-s + (0.155 + 0.0898i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.261 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.261 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.261 - 0.965i$
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: \((2,\ 2736,\ (\ :1/2),\ 0.261 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.871926711\)
\(L(\frac12)\) \(\approx\) \(1.871926711\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.88 + 3.92i)T \)
good5 \( 1 + (0.406 - 0.234i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.49T + 7T^{2} \)
11 \( 1 - 1.34iT - 11T^{2} \)
13 \( 1 + (-4.30 - 2.48i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (6.31 - 3.64i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-6.97 - 4.02i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.74 - 8.21i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.95iT - 31T^{2} \)
37 \( 1 + 0.581iT - 37T^{2} \)
41 \( 1 + (0.761 + 1.31i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.63 - 8.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.06 - 0.615i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.09 + 8.82i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.09 + 8.82i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.59 - 6.22i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.23 + 5.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.67 - 6.36i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.703 + 1.21i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.56 + 0.903i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.53iT - 83T^{2} \)
89 \( 1 + (7.73 - 13.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.66 + 5.58i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.892643067630526632561836791137, −8.367220001501640954106272858440, −7.38873528242192138911722501262, −6.85143422155157877986746227124, −5.90146407803890450622137192057, −4.94461508337981232980245587779, −4.33237302631160930806305666918, −3.46471044812499878147562762967, −2.07754233013046839908947270618, −1.38776118228057043009745855472, 0.62925634432239205001038068228, 1.83886990528796772139584905646, 2.85763023995080347436247817850, 4.08988745291041491787443751779, 4.61046774489899792900817629059, 5.60260876167643186712313873590, 6.27657366626725985949415172295, 7.28581886227288884551323358841, 8.022349211407386491831167414956, 8.663243293466897166297553342302

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