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Properties

Label 2-3784-3784.1659-c0-0-0
Degree $2$
Conductor $3784$
Sign $-0.372 - 0.928i$
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  + (−0.691 + 0.722i)2-s + (1.10 − 0.0329i)3-s + (−0.0448 − 0.998i)4-s + (−0.737 + 0.818i)6-s + (0.753 + 0.657i)8-s + (0.213 − 0.0127i)9-s + (−0.999 + 0.0299i)11-s + (−0.0823 − 1.09i)12-s + (−0.995 + 0.0896i)16-s + (−0.708 + 1.52i)17-s + (−0.138 + 0.163i)18-s + (0.446 + 1.71i)19-s + (0.669 − 0.743i)22-s + (0.851 + 0.699i)24-s + (0.525 + 0.850i)25-s + ⋯
L(s)  = 1  + (−0.691 + 0.722i)2-s + (1.10 − 0.0329i)3-s + (−0.0448 − 0.998i)4-s + (−0.737 + 0.818i)6-s + (0.753 + 0.657i)8-s + (0.213 − 0.0127i)9-s + (−0.999 + 0.0299i)11-s + (−0.0823 − 1.09i)12-s + (−0.995 + 0.0896i)16-s + (−0.708 + 1.52i)17-s + (−0.138 + 0.163i)18-s + (0.446 + 1.71i)19-s + (0.669 − 0.743i)22-s + (0.851 + 0.699i)24-s + (0.525 + 0.850i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3784\)    =    \(2^{3} \cdot 11 \cdot 43\)
Sign: $-0.372 - 0.928i$
Analytic conductor: \(1.88846\)
Root analytic conductor: \(1.37421\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3784} (1659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3784,\ (\ :0),\ -0.372 - 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.023386524\)
\(L(\frac12)\) \(\approx\) \(1.023386524\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.691 - 0.722i)T \)
11 \( 1 + (0.999 - 0.0299i)T \)
43 \( 1 + (-0.791 - 0.611i)T \)
good3 \( 1 + (-1.10 + 0.0329i)T + (0.998 - 0.0598i)T^{2} \)
5 \( 1 + (-0.525 - 0.850i)T^{2} \)
7 \( 1 + (-0.669 + 0.743i)T^{2} \)
13 \( 1 + (0.337 + 0.941i)T^{2} \)
17 \( 1 + (0.708 - 1.52i)T + (-0.646 - 0.762i)T^{2} \)
19 \( 1 + (-0.446 - 1.71i)T + (-0.873 + 0.486i)T^{2} \)
23 \( 1 + (-0.955 - 0.294i)T^{2} \)
29 \( 1 + (-0.998 - 0.0598i)T^{2} \)
31 \( 1 + (-0.887 - 0.460i)T^{2} \)
37 \( 1 + (0.978 + 0.207i)T^{2} \)
41 \( 1 + (-0.0294 + 0.00534i)T + (0.936 - 0.351i)T^{2} \)
47 \( 1 + (-0.858 - 0.512i)T^{2} \)
53 \( 1 + (0.999 + 0.0299i)T^{2} \)
59 \( 1 + (0.508 - 0.443i)T + (0.134 - 0.990i)T^{2} \)
61 \( 1 + (-0.887 + 0.460i)T^{2} \)
67 \( 1 + (0.0763 - 0.194i)T + (-0.733 - 0.680i)T^{2} \)
71 \( 1 + (0.599 - 0.800i)T^{2} \)
73 \( 1 + (-1.17 - 1.56i)T + (-0.280 + 0.959i)T^{2} \)
79 \( 1 + (0.104 - 0.994i)T^{2} \)
83 \( 1 + (-0.387 + 1.32i)T + (-0.842 - 0.538i)T^{2} \)
89 \( 1 + (-0.951 - 0.648i)T + (0.365 + 0.930i)T^{2} \)
97 \( 1 + (0.871 + 1.32i)T + (-0.393 + 0.919i)T^{2} \)
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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

−8.712224494957133524647572256119, −8.128429559759654192606561142468, −7.76574575080916290611561251683, −6.94675902566566462413018858384, −5.93777578898398415415898076262, −5.47544531551250338677865929157, −4.31555577056714582613320585954, −3.42371814014184112140520138396, −2.34566450918829143386508284901, −1.53160605594930521607689722158, 0.61952235013291212279426804108, 2.37600644764126645940673149224, 2.58640047681038057615463816484, 3.38562668949107393676720550850, 4.50302106740403451487000200275, 5.14499438693921805323013505246, 6.58437801129100033765836089360, 7.37598896759214702330784347354, 7.83836809372383563939344656037, 8.675274145240154247634373069000

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