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Properties

Label 2-2736-19.7-c1-0-7
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  + (−2 − 3.46i)5-s + 3·7-s − 4·11-s + (−2.5 + 4.33i)13-s + (4 + 1.73i)19-s + (2 − 3.46i)23-s + (−5.49 + 9.52i)25-s + (−4 + 6.92i)29-s − 31-s + (−6 − 10.3i)35-s − 5·37-s + (−4 − 6.92i)41-s + (−2.5 − 4.33i)43-s + (−4 + 6.92i)47-s + 2·49-s + ⋯
L(s)  = 1  + (−0.894 − 1.54i)5-s + 1.13·7-s − 1.20·11-s + (−0.693 + 1.20i)13-s + (0.917 + 0.397i)19-s + (0.417 − 0.722i)23-s + (−1.09 + 1.90i)25-s + (−0.742 + 1.28i)29-s − 0.179·31-s + (−1.01 − 1.75i)35-s − 0.821·37-s + (−0.624 − 1.08i)41-s + (−0.381 − 0.660i)43-s + (−0.583 + 1.01i)47-s + 0.285·49-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: $\chi_{2736} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.0977 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6798430273\)
\(L(\frac12)\) \(\approx\) \(0.6798430273\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + 3.46i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
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 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4 - 6.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 + (4 + 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8 - 13.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.5 - 12.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.5 + 6.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
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

−8.746604429847896594359142399062, −8.379771143879936210409614059627, −7.54583335170704073171108238188, −7.09551689925654180195612114733, −5.45098654943333966895459852095, −5.06493577267858855801850465563, −4.50116185918283351293819646895, −3.56781055482673680340701383548, −2.12309838856867539168310622014, −1.17250722444305766221799270435, 0.23297870543730096797677302709, 2.08450207806930729844466128052, 3.00173730888348953393402973878, 3.57755984016781084058066171680, 4.92027892878213549974287614787, 5.32788757600972652638742002583, 6.53259120101027644689625391772, 7.37187026073386030937008334899, 7.931069722793744539918073359351, 8.092798621716948607595969057627

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