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Properties

Label 2-3784-3784.955-c0-0-0
Degree $2$
Conductor $3784$
Sign $0.0216 + 0.999i$
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  + (0.936 − 0.351i)2-s + (−1.76 + 0.917i)3-s + (0.753 − 0.657i)4-s + (−1.33 + 1.48i)6-s + (0.473 − 0.880i)8-s + (1.70 − 2.42i)9-s + (0.887 − 0.460i)11-s + (−0.727 + 1.85i)12-s + (0.134 − 0.990i)16-s + (−1.49 − 1.15i)17-s + (0.746 − 2.87i)18-s + (−0.231 + 0.309i)19-s + (0.669 − 0.743i)22-s + (−0.0297 + 1.99i)24-s + (−0.842 − 0.538i)25-s + ⋯
L(s)  = 1  + (0.936 − 0.351i)2-s + (−1.76 + 0.917i)3-s + (0.753 − 0.657i)4-s + (−1.33 + 1.48i)6-s + (0.473 − 0.880i)8-s + (1.70 − 2.42i)9-s + (0.887 − 0.460i)11-s + (−0.727 + 1.85i)12-s + (0.134 − 0.990i)16-s + (−1.49 − 1.15i)17-s + (0.746 − 2.87i)18-s + (−0.231 + 0.309i)19-s + (0.669 − 0.743i)22-s + (−0.0297 + 1.99i)24-s + (−0.842 − 0.538i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0216 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0216 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3784\)    =    \(2^{3} \cdot 11 \cdot 43\)
Sign: $0.0216 + 0.999i$
Analytic conductor: \(1.88846\)
Root analytic conductor: \(1.37421\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3784} (955, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3784,\ (\ :0),\ 0.0216 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.061903673\)
\(L(\frac12)\) \(\approx\) \(1.061903673\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.936 + 0.351i)T \)
11 \( 1 + (-0.887 + 0.460i)T \)
43 \( 1 + (0.447 + 0.894i)T \)
good3 \( 1 + (1.76 - 0.917i)T + (0.575 - 0.817i)T^{2} \)
5 \( 1 + (0.842 + 0.538i)T^{2} \)
7 \( 1 + (-0.669 + 0.743i)T^{2} \)
13 \( 1 + (-0.712 - 0.701i)T^{2} \)
17 \( 1 + (1.49 + 1.15i)T + (0.251 + 0.967i)T^{2} \)
19 \( 1 + (0.231 - 0.309i)T + (-0.280 - 0.959i)T^{2} \)
23 \( 1 + (-0.0747 + 0.997i)T^{2} \)
29 \( 1 + (-0.575 - 0.817i)T^{2} \)
31 \( 1 + (-0.193 - 0.981i)T^{2} \)
37 \( 1 + (0.978 + 0.207i)T^{2} \)
41 \( 1 + (1.87 + 0.516i)T + (0.858 + 0.512i)T^{2} \)
47 \( 1 + (0.691 - 0.722i)T^{2} \)
53 \( 1 + (-0.887 - 0.460i)T^{2} \)
59 \( 1 + (-0.675 - 1.25i)T + (-0.550 + 0.834i)T^{2} \)
61 \( 1 + (-0.193 + 0.981i)T^{2} \)
67 \( 1 + (0.199 - 0.0616i)T + (0.826 - 0.563i)T^{2} \)
71 \( 1 + (0.646 + 0.762i)T^{2} \)
73 \( 1 + (-1.26 + 1.49i)T + (-0.163 - 0.986i)T^{2} \)
79 \( 1 + (0.104 - 0.994i)T^{2} \)
83 \( 1 + (0.306 + 1.84i)T + (-0.946 + 0.323i)T^{2} \)
89 \( 1 + (-1.82 - 0.275i)T + (0.955 + 0.294i)T^{2} \)
97 \( 1 + (-0.891 - 0.0801i)T + (0.983 + 0.178i)T^{2} \)
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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

−8.853076852071117272768094956005, −7.21479712299058992101132238453, −6.60975849929278636841060994835, −6.15282728094547595886163291974, −5.33617023566654572746255366856, −4.76805279775248198866122645607, −4.05853880091288651075014613815, −3.46538091707352780132003234657, −1.93933521214055025403977541102, −0.54917114918491496339735544751, 1.53351010357297134972497544461, 2.18141420811670117394232599787, 3.85758356529228055060422254428, 4.54259844904923223298003986634, 5.20504737547987712977788333880, 6.01834313934254871810130812705, 6.64338324350306107601757552842, 6.83531776808254547741685072010, 7.78686732388335798943536278146, 8.495940513103808118926737215329

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