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Properties

Label 2-3789-421.298-c0-0-0
Degree $2$
Conductor $3789$
Sign $0.965 - 0.262i$
Analytic cond. $1.89095$
Root an. cond. $1.37512$
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  + (−0.351 − 0.936i)4-s + (−0.0714 + 0.258i)7-s + (1.54 + 0.789i)13-s + (−0.753 + 0.657i)16-s + (−1.02 + 1.34i)19-s + (0.753 + 0.657i)25-s + (0.267 − 0.0240i)28-s + (0.433 + 1.01i)31-s + (0.296 − 0.270i)37-s + (0.223 + 0.0150i)43-s + (0.796 + 0.475i)49-s + (0.194 − 1.72i)52-s + (0.0993 − 1.47i)61-s + (0.880 + 0.473i)64-s + (−1.37 − 0.446i)67-s + ⋯
L(s)  = 1  + (−0.351 − 0.936i)4-s + (−0.0714 + 0.258i)7-s + (1.54 + 0.789i)13-s + (−0.753 + 0.657i)16-s + (−1.02 + 1.34i)19-s + (0.753 + 0.657i)25-s + (0.267 − 0.0240i)28-s + (0.433 + 1.01i)31-s + (0.296 − 0.270i)37-s + (0.223 + 0.0150i)43-s + (0.796 + 0.475i)49-s + (0.194 − 1.72i)52-s + (0.0993 − 1.47i)61-s + (0.880 + 0.473i)64-s + (−1.37 − 0.446i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3789 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3789 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3789\)    =    \(3^{2} \cdot 421\)
Sign: $0.965 - 0.262i$
Analytic conductor: \(1.89095\)
Root analytic conductor: \(1.37512\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3789} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3789,\ (\ :0),\ 0.965 - 0.262i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.177058302\)
\(L(\frac12)\) \(\approx\) \(1.177058302\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
421 \( 1 + (-0.266 - 0.963i)T \)
good2 \( 1 + (0.351 + 0.936i)T^{2} \)
5 \( 1 + (-0.753 - 0.657i)T^{2} \)
7 \( 1 + (0.0714 - 0.258i)T + (-0.858 - 0.512i)T^{2} \)
11 \( 1 + (-0.995 - 0.0896i)T^{2} \)
13 \( 1 + (-1.54 - 0.789i)T + (0.587 + 0.809i)T^{2} \)
17 \( 1 + (0.550 + 0.834i)T^{2} \)
19 \( 1 + (1.02 - 1.34i)T + (-0.266 - 0.963i)T^{2} \)
23 \( 1 + (0.919 + 0.393i)T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + (-0.433 - 1.01i)T + (-0.691 + 0.722i)T^{2} \)
37 \( 1 + (-0.296 + 0.270i)T + (0.0896 - 0.995i)T^{2} \)
41 \( 1 + (0.266 + 0.963i)T^{2} \)
43 \( 1 + (-0.223 - 0.0150i)T + (0.990 + 0.134i)T^{2} \)
47 \( 1 + (-0.657 + 0.753i)T^{2} \)
53 \( 1 + (-0.512 + 0.858i)T^{2} \)
59 \( 1 + (-0.998 + 0.0448i)T^{2} \)
61 \( 1 + (-0.0993 + 1.47i)T + (-0.990 - 0.134i)T^{2} \)
67 \( 1 + (1.37 + 0.446i)T + (0.809 + 0.587i)T^{2} \)
71 \( 1 + (-0.998 - 0.0448i)T^{2} \)
73 \( 1 + (-0.309 - 0.0489i)T + (0.951 + 0.309i)T^{2} \)
79 \( 1 + (0.268 + 0.234i)T + (0.134 + 0.990i)T^{2} \)
83 \( 1 + (0.998 - 0.0448i)T^{2} \)
89 \( 1 + (0.722 - 0.691i)T^{2} \)
97 \( 1 + (-0.0240 + 0.177i)T + (-0.963 - 0.266i)T^{2} \)
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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

−8.783017177708398778688808742095, −8.233592474887645656815544860881, −7.08787061547849333901193118009, −6.20276725304416521393178143353, −5.97611105457425844885409843705, −4.94595561122994738510653060710, −4.17941598108240883724671973242, −3.38052110526322327033114338099, −1.99144209489357677557409930530, −1.22812828783121875635892380257, 0.789926711516203574600745844179, 2.43352049523648017554539635897, 3.18373066817819044114710319355, 4.09248525094302740281921168401, 4.60799918093076909281021951893, 5.73931328519788592244190347230, 6.50630701485012434181985428088, 7.23292238970571606930718981003, 8.049751755338652914627259490164, 8.649702979453702760010601915394

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