Properties

Label 2-2736-76.23-c0-0-0
Degree $2$
Conductor $2736$
Sign $-0.305 - 0.952i$
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  + (−1.70 − 0.984i)7-s + (−0.266 + 1.50i)13-s + (−0.5 + 0.866i)19-s + (−0.173 + 0.984i)25-s + (0.592 + 0.342i)31-s − 1.87·37-s + (0.439 + 0.524i)43-s + (1.43 + 2.49i)49-s + (1.17 + 0.984i)61-s + (0.673 + 1.85i)67-s + (−0.266 − 1.50i)73-s + (−1.93 + 0.342i)79-s + (1.93 − 2.31i)91-s + (−0.939 − 0.342i)97-s + (−1.11 + 0.642i)103-s + ⋯
L(s)  = 1  + (−1.70 − 0.984i)7-s + (−0.266 + 1.50i)13-s + (−0.5 + 0.866i)19-s + (−0.173 + 0.984i)25-s + (0.592 + 0.342i)31-s − 1.87·37-s + (0.439 + 0.524i)43-s + (1.43 + 2.49i)49-s + (1.17 + 0.984i)61-s + (0.673 + 1.85i)67-s + (−0.266 − 1.50i)73-s + (−1.93 + 0.342i)79-s + (1.93 − 2.31i)91-s + (−0.939 − 0.342i)97-s + (−1.11 + 0.642i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.305 - 0.952i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :0),\ -0.305 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5272807407\)
\(L(\frac12)\) \(\approx\) \(0.5272807407\)
\(L(1)\) 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 \( 1 + (0.173 - 0.984i)T^{2} \)
7 \( 1 + (1.70 + 0.984i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \)
17 \( 1 + (0.766 + 0.642i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (0.766 - 0.642i)T^{2} \)
31 \( 1 + (-0.592 - 0.342i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + 1.87T + T^{2} \)
41 \( 1 + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.439 - 0.524i)T + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + (0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
67 \( 1 + (-0.673 - 1.85i)T + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (1.93 - 0.342i)T + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
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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

−9.278988447776399316546325451914, −8.657288753045057223104346096671, −7.47909079964892840427089321788, −6.88523088785183660164639558418, −6.42091143486856957842009071032, −5.46842488632759760847589422534, −4.22458089905394565061156896591, −3.77109988308146855970313278162, −2.79863134501186826309233371785, −1.48433857041866618193387588187, 0.32742209667497723651042750906, 2.35739860201720531077296559279, 2.96589084186042026135142432250, 3.79994411513703291413319344133, 5.07471295752938278995300378427, 5.74303968164002686555072796897, 6.47524183384080472555806971250, 7.09385269860957308973443972037, 8.223691025311362736664985112590, 8.750281116234435246097977540894

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