Properties

Label 2-1368-1368.187-c0-0-1
Degree $2$
Conductor $1368$
Sign $0.188 + 0.982i$
Analytic cond. $0.682720$
Root an. cond. $0.826269$
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  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)3-s + (0.766 − 0.642i)4-s + (−0.5 + 0.866i)6-s + (−0.500 + 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.766 − 1.32i)11-s + (0.173 − 0.984i)12-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + (0.173 + 0.984i)18-s + (−0.939 − 0.342i)19-s + (1.17 + 0.984i)22-s + (0.173 + 0.984i)24-s + (0.766 − 0.642i)25-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)3-s + (0.766 − 0.642i)4-s + (−0.5 + 0.866i)6-s + (−0.500 + 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.766 − 1.32i)11-s + (0.173 − 0.984i)12-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + (0.173 + 0.984i)18-s + (−0.939 − 0.342i)19-s + (1.17 + 0.984i)22-s + (0.173 + 0.984i)24-s + (0.766 − 0.642i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.188 + 0.982i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.188 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8087379840\)
\(L(\frac12)\) \(\approx\) \(0.8087379840\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 - 0.342i)T \)
3 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (0.939 + 0.342i)T \)
good5 \( 1 + (-0.766 + 0.642i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.939 - 0.342i)T^{2} \)
17 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
43 \( 1 + (-1.53 - 1.28i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (0.939 - 0.342i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 - 0.347T + T^{2} \)
89 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
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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

−9.169333400745475761564380857726, −8.836920787296374297964417210058, −8.033505694401821699922406230046, −7.43331569231267615829307100097, −6.45791984511241320261166123584, −5.92124389340760830684436596547, −4.55211341208655414162086142303, −2.98005393032754665986998964040, −2.37030091056624206715985349479, −0.816541839630491260965869587170, 1.93817065188632081708200336201, 2.58776348699621325958003745138, 3.85014262358980424989852617003, 4.59808224015207376683219126202, 5.91742472623838857256860854639, 7.24502468041259360662276318661, 7.56736019353816724831494730967, 8.769636059091292721164812095272, 8.948046567713926529798551226215, 10.03207380736555093772018079819

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