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Properties

Label 2-2736-19.7-c1-0-15
Degree $2$
Conductor $2736$
Sign $-0.412 - 0.910i$
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  + (1.95 + 3.39i)5-s − 2.04·7-s + 1.91·11-s + (2.13 − 3.68i)13-s + (3.04 + 5.27i)17-s + (−4.33 + 0.497i)19-s + (−3.42 + 5.93i)23-s + (−5.17 + 8.96i)25-s + (1.34 − 2.32i)29-s + 1.46·31-s + (−4.00 − 6.93i)35-s + 3.93·37-s + (1.70 + 2.94i)41-s + (3.77 + 6.54i)43-s + (4.96 − 8.59i)47-s + ⋯
L(s)  = 1  + (0.875 + 1.51i)5-s − 0.772·7-s + 0.578·11-s + (0.590 − 1.02i)13-s + (0.738 + 1.27i)17-s + (−0.993 + 0.114i)19-s + (−0.714 + 1.23i)23-s + (−1.03 + 1.79i)25-s + (0.249 − 0.431i)29-s + 0.263·31-s + (−0.676 − 1.17i)35-s + 0.647·37-s + (0.265 + 0.459i)41-s + (0.576 + 0.997i)43-s + (0.723 − 1.25i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.412 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.412 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.412 - 0.910i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.412 - 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.775117362\)
\(L(\frac12)\) \(\approx\) \(1.775117362\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.33 - 0.497i)T \)
good5 \( 1 + (-1.95 - 3.39i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 2.04T + 7T^{2} \)
11 \( 1 - 1.91T + 11T^{2} \)
13 \( 1 + (-2.13 + 3.68i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.04 - 5.27i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.42 - 5.93i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.34 + 2.32i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.46T + 31T^{2} \)
37 \( 1 - 3.93T + 37T^{2} \)
41 \( 1 + (-1.70 - 2.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.77 - 6.54i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.96 + 8.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.91 - 8.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.61 + 2.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.70 - 8.14i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.28 + 9.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.21 - 3.84i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.33 + 10.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.52 - 11.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.43T + 83T^{2} \)
89 \( 1 + (-3.47 + 6.01i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.58 - 4.48i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.318934268848540720962441044858, −8.157722679423213625806320087568, −7.55931673945483650100982756616, −6.40464912904802108620702099002, −6.23058982430966562586883080080, −5.60445465465908165020653448774, −4.00172744913731884860912719783, −3.36574664489894154256401333243, −2.57338859361634226239426696525, −1.46206129885887239321561766997, 0.57657016331313014779934720418, 1.63952661660482051064669965990, 2.66228445622116124099603020707, 4.03280407900874874897988622673, 4.59282644634279539048162649329, 5.52846029295187988195593973708, 6.27878189595192760030032001243, 6.81509102851897199766061169319, 8.062628858559658234643451290488, 8.794883509218863751743749846652

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