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Properties

Label 2-3784-3784.267-c0-0-1
Degree $2$
Conductor $3784$
Sign $-0.779 - 0.625i$
Analytic cond. $1.88846$
Root an. cond. $1.37421$
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  + (−0.963 − 0.266i)2-s + (−0.246 − 1.48i)3-s + (0.858 + 0.512i)4-s + (−0.157 + 1.49i)6-s + (−0.691 − 0.722i)8-s + (−1.20 + 0.409i)9-s + (−0.163 − 0.986i)11-s + (0.550 − 1.40i)12-s + (0.473 + 0.880i)16-s + (−0.0299 − 0.000895i)17-s + (1.26 − 0.0758i)18-s + (−1.00 − 1.42i)19-s + (−0.104 + 0.994i)22-s + (−0.903 + 1.20i)24-s + (−0.772 + 0.635i)25-s + ⋯
L(s)  = 1  + (−0.963 − 0.266i)2-s + (−0.246 − 1.48i)3-s + (0.858 + 0.512i)4-s + (−0.157 + 1.49i)6-s + (−0.691 − 0.722i)8-s + (−1.20 + 0.409i)9-s + (−0.163 − 0.986i)11-s + (0.550 − 1.40i)12-s + (0.473 + 0.880i)16-s + (−0.0299 − 0.000895i)17-s + (1.26 − 0.0758i)18-s + (−1.00 − 1.42i)19-s + (−0.104 + 0.994i)22-s + (−0.903 + 1.20i)24-s + (−0.772 + 0.635i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3784\)    =    \(2^{3} \cdot 11 \cdot 43\)
Sign: $-0.779 - 0.625i$
Analytic conductor: \(1.88846\)
Root analytic conductor: \(1.37421\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3784} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3784,\ (\ :0),\ -0.779 - 0.625i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4094731045\)
\(L(\frac12)\) \(\approx\) \(0.4094731045\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.963 + 0.266i)T \)
11 \( 1 + (0.163 + 0.986i)T \)
43 \( 1 + (-0.887 - 0.460i)T \)
good3 \( 1 + (0.246 + 1.48i)T + (-0.946 + 0.323i)T^{2} \)
5 \( 1 + (0.772 - 0.635i)T^{2} \)
7 \( 1 + (0.104 - 0.994i)T^{2} \)
13 \( 1 + (0.447 - 0.894i)T^{2} \)
17 \( 1 + (0.0299 + 0.000895i)T + (0.998 + 0.0598i)T^{2} \)
19 \( 1 + (1.00 + 1.42i)T + (-0.337 + 0.941i)T^{2} \)
23 \( 1 + (-0.0747 + 0.997i)T^{2} \)
29 \( 1 + (0.946 + 0.323i)T^{2} \)
31 \( 1 + (0.873 - 0.486i)T^{2} \)
37 \( 1 + (-0.913 + 0.406i)T^{2} \)
41 \( 1 + (-0.712 + 1.07i)T + (-0.393 - 0.919i)T^{2} \)
47 \( 1 + (-0.983 + 0.178i)T^{2} \)
53 \( 1 + (0.163 - 0.986i)T^{2} \)
59 \( 1 + (-0.618 + 0.646i)T + (-0.0448 - 0.998i)T^{2} \)
61 \( 1 + (0.873 + 0.486i)T^{2} \)
67 \( 1 + (1.86 - 0.576i)T + (0.826 - 0.563i)T^{2} \)
71 \( 1 + (0.925 - 0.379i)T^{2} \)
73 \( 1 + (1.69 + 0.692i)T + (0.712 + 0.701i)T^{2} \)
79 \( 1 + (0.978 - 0.207i)T^{2} \)
83 \( 1 + (1.37 + 1.35i)T + (0.0149 + 0.999i)T^{2} \)
89 \( 1 + (1.03 + 0.156i)T + (0.955 + 0.294i)T^{2} \)
97 \( 1 + (-1.33 - 1.16i)T + (0.134 + 0.990i)T^{2} \)
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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

−8.236884064177703337193294829521, −7.42320518466172061469944534357, −7.07560305298205044183116736152, −6.14382476362813751367682695590, −5.77634128857311481682420967904, −4.29517969440754478096201585829, −3.04816562277375520033361432026, −2.34305256490097797820746732185, −1.39249449002814983256609388244, −0.32765444993406471542448969041, 1.71367498825979228810091706693, 2.74771276549136747063431060401, 3.96556010617327840694289676121, 4.51018891849423642987107373428, 5.55066675610018600128709851682, 6.04700566691639864182718862575, 7.02120094691422000937653619531, 7.85190888159322772896628748047, 8.517529370602273386988712760059, 9.259351715221728135628747905781

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