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Properties

Label 2-2736-76.31-c1-0-3
Degree $2$
Conductor $2736$
Sign $-0.977 - 0.211i$
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 + 4.56i·7-s + 5.65i·11-s + (3.94 + 2.28i)13-s + (−1 − 1.73i)17-s + (−4 + 1.73i)19-s + (−4.89 − 2.82i)23-s + (0.500 − 0.866i)25-s + (1.89 + 1.09i)29-s − 7.89·31-s + (7.89 − 4.56i)35-s − 1.09i·37-s + (−4.5 + 2.59i)43-s + (7.89 + 4.56i)47-s − 13.7·49-s + ⋯
L(s)  = 1  + (−0.447 − 0.774i)5-s + 1.72i·7-s + 1.70i·11-s + (1.09 + 0.632i)13-s + (−0.242 − 0.420i)17-s + (−0.917 + 0.397i)19-s + (−1.02 − 0.589i)23-s + (0.100 − 0.173i)25-s + (0.352 + 0.203i)29-s − 1.41·31-s + (1.33 − 0.770i)35-s − 0.180i·37-s + (−0.686 + 0.396i)43-s + (1.15 + 0.665i)47-s − 1.97·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.977 - 0.211i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1855, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.977 - 0.211i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6676593177\)
\(L(\frac12)\) \(\approx\) \(0.6676593177\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.56iT - 7T^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 + (-3.94 - 2.28i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (4.89 + 2.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.89 - 1.09i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.89T + 31T^{2} \)
37 \( 1 + 1.09iT - 37T^{2} \)
41 \( 1 + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.5 - 2.59i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.89 - 4.56i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.8 + 6.29i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.89 + 10.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.94 + 8.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.39 + 7.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2 - 3.46i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.39 - 5.88i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.0505 - 0.0874i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + (-1.89 - 1.09i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.79 + 2.19i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.220246369511728966932876863709, −8.393247531561273685018714339192, −7.978269197843624607617423961819, −6.72547360916542750894872937601, −6.18555486886294589240016920299, −5.15912975654312876832663793728, −4.57911619911788764171015984535, −3.72326233336191596975280139638, −2.30080454270453657724602016803, −1.75523568263743574733326806083, 0.21721572510602863287202598633, 1.34521071993339748236734242434, 2.98171767610243353027852636744, 3.74900312200181241021267056200, 4.08007133134033340732390577076, 5.54357393070144678616009120920, 6.27317117144062370610258299569, 6.96130919876948927644716162218, 7.74771729547796181725441129742, 8.310443066701506375592181162299

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