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Properties

Label 2-2736-19.11-c1-0-30
Degree $2$
Conductor $2736$
Sign $0.856 + 0.516i$
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  + (1.22 − 2.12i)5-s + 3.44·7-s − 2.44·11-s + (0.5 + 0.866i)13-s + (2.44 − 4.24i)17-s + (−1 + 4.24i)19-s + (4.22 + 7.31i)23-s + (−0.499 − 0.866i)25-s + (−2.44 − 4.24i)29-s + 9.44·31-s + (4.22 − 7.31i)35-s + 8.79·37-s + (−1.72 + 2.98i)43-s + (−0.550 − 0.953i)47-s + 4.89·49-s + ⋯
L(s)  = 1  + (0.547 − 0.948i)5-s + 1.30·7-s − 0.738·11-s + (0.138 + 0.240i)13-s + (0.594 − 1.02i)17-s + (−0.229 + 0.973i)19-s + (0.880 + 1.52i)23-s + (−0.0999 − 0.173i)25-s + (−0.454 − 0.787i)29-s + 1.69·31-s + (0.714 − 1.23i)35-s + 1.44·37-s + (−0.263 + 0.455i)43-s + (−0.0803 − 0.139i)47-s + 0.699·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.856 + 0.516i$
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: \((2,\ 2736,\ (\ :1/2),\ 0.856 + 0.516i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.458822261\)
\(L(\frac12)\) \(\approx\) \(2.458822261\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 4.24i)T \)
good5 \( 1 + (-1.22 + 2.12i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.44T + 7T^{2} \)
11 \( 1 + 2.44T + 11T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.44 + 4.24i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-4.22 - 7.31i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.44 + 4.24i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.44T + 31T^{2} \)
37 \( 1 - 8.79T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.72 - 2.98i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.550 + 0.953i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.22 + 2.12i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.67 + 6.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.05 - 1.81i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.27 + 3.94i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.94 + 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.17 + 10.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.8T + 83T^{2} \)
89 \( 1 + (1.77 + 3.07i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.34 - 9.26i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.716980215565592772090333540640, −7.88856000411421651512759374294, −7.63616320866004572568662610331, −6.32070748877121970874154005619, −5.37297991203305265203204394910, −5.05447872020811688752226751843, −4.21436233724926755625181091434, −2.93445201699375731243569214768, −1.80166654379551554099750414297, −1.00590594728290876301023933007, 1.10817947382482915922531723034, 2.37891410509404473329934743639, 2.90662205789411750135157032890, 4.28523323120829915625205039347, 4.98032411430193180689358283992, 5.84230816844695744911190660133, 6.61923573860718157432374332215, 7.35436186404268614104283222111, 8.279570364450346839117818041377, 8.589584729363780745735859616911

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