Properties

Label 2-2736-76.35-c0-0-0
Degree $2$
Conductor $2736$
Sign $0.756 - 0.654i$
Analytic cond. $1.36544$
Root an. cond. $1.16852$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.592 + 0.342i)7-s + (0.326 + 0.118i)13-s + (0.5 + 0.866i)19-s + (0.939 + 0.342i)25-s + (−1.11 + 0.642i)31-s + 1.53·37-s + (1.26 + 0.223i)43-s + (−0.266 + 0.460i)49-s + (0.0603 + 0.342i)61-s + (0.439 + 0.524i)67-s + (0.326 − 0.118i)73-s + (0.233 + 0.642i)79-s + (−0.233 + 0.0412i)91-s + (0.766 + 0.642i)97-s + (−1.70 − 0.984i)103-s + ⋯
L(s)  = 1  + (−0.592 + 0.342i)7-s + (0.326 + 0.118i)13-s + (0.5 + 0.866i)19-s + (0.939 + 0.342i)25-s + (−1.11 + 0.642i)31-s + 1.53·37-s + (1.26 + 0.223i)43-s + (−0.266 + 0.460i)49-s + (0.0603 + 0.342i)61-s + (0.439 + 0.524i)67-s + (0.326 − 0.118i)73-s + (0.233 + 0.642i)79-s + (−0.233 + 0.0412i)91-s + (0.766 + 0.642i)97-s + (−1.70 − 0.984i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.756 - 0.654i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :0),\ 0.756 - 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.137507613\)
\(L(\frac12)\) \(\approx\) \(1.137507613\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.939 - 0.342i)T^{2} \)
7 \( 1 + (0.592 - 0.342i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \)
17 \( 1 + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (0.173 - 0.984i)T^{2} \)
31 \( 1 + (1.11 - 0.642i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 - 1.53T + T^{2} \)
41 \( 1 + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (-1.26 - 0.223i)T + (0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (-0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.439 - 0.524i)T + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (-0.233 - 0.642i)T + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \)
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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

−9.222837292315297051375742273993, −8.362430086959947009880632574405, −7.58144956104003611126479901307, −6.79759073898010487548091335938, −5.99078759114862029209654087002, −5.36494422859218111114762113446, −4.29419107931578137219567759459, −3.42187450396559928580983996589, −2.57761491931209092971974309877, −1.27922242948173398644991721308, 0.836938239494087750550743392080, 2.33910994776101595092448930334, 3.25943900846408191559638608243, 4.11859874466801241384100931819, 5.02085762841085533229893890530, 5.92130306300428571782473227279, 6.66486898434618355981375049220, 7.37724814627203980488948904250, 8.114230768003278819796456581509, 9.107420199735405275671523380291

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