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Properties

Label 2-2736-19.11-c1-0-13
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 + 5·7-s − 4·11-s + (−2.5 − 4.33i)13-s + (−4 + 1.73i)19-s + (3 + 5.19i)23-s + (0.500 + 0.866i)25-s + (4 + 6.92i)29-s + 31-s + (−5 + 8.66i)35-s + 7·37-s + (−5.5 + 9.52i)43-s + (−5 − 8.66i)47-s + 18·49-s + (3 + 5.19i)53-s + ⋯
L(s)  = 1  + (−0.447 + 0.774i)5-s + 1.88·7-s − 1.20·11-s + (−0.693 − 1.20i)13-s + (−0.917 + 0.397i)19-s + (0.625 + 1.08i)23-s + (0.100 + 0.173i)25-s + (0.742 + 1.28i)29-s + 0.179·31-s + (−0.845 + 1.46i)35-s + 1.15·37-s + (−0.838 + 1.45i)43-s + (−0.729 − 1.26i)47-s + 2.57·49-s + (0.412 + 0.713i)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: $\chi_{2736} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.0977 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.496693639\)
\(L(\frac12)\) \(\approx\) \(1.496693639\)
\(L(\frac{3}{2})\) 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 \( 1 - 5T + 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4 - 6.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 - 7T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.5 - 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5 + 8.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.5 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (6 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.706352139263630936044012110245, −8.136805667153587877195395437590, −7.59322712672205447095886873809, −7.06707039432137499667419833894, −5.73366441506948604091137256453, −5.10547977400353283724261130863, −4.48311017739948900250897140516, −3.21061298426581242274874067147, −2.49648583472268884474435854334, −1.28703331155673107417395809003, 0.50021097979098971548336206999, 1.87338182415055904577700401789, 2.58112772858305102947161695747, 4.36692886954130056017159836809, 4.59282919297359213102333104112, 5.16114020840914364488774191536, 6.35586218098762543330109394937, 7.30712905044490937376951120611, 8.144241001324091353134128451947, 8.343676052850351411112881621887

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