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Properties

Label 2-3789-421.42-c0-0-0
Degree $2$
Conductor $3789$
Sign $-0.187 + 0.982i$
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  + (−0.512 + 0.858i)4-s + (−1.01 + 0.433i)7-s + (0.338 − 0.663i)13-s + (−0.473 − 0.880i)16-s + (−1.84 − 0.378i)19-s + (0.473 − 0.880i)25-s + (0.147 − 1.09i)28-s + (−1.95 + 0.355i)31-s + (1.53 − 0.103i)37-s + (−1.01 − 0.306i)43-s + (0.147 − 0.154i)49-s + (0.396 + 0.630i)52-s + (0.573 − 1.91i)61-s + (0.998 + 0.0448i)64-s + (0.668 + 0.217i)67-s + ⋯
L(s)  = 1  + (−0.512 + 0.858i)4-s + (−1.01 + 0.433i)7-s + (0.338 − 0.663i)13-s + (−0.473 − 0.880i)16-s + (−1.84 − 0.378i)19-s + (0.473 − 0.880i)25-s + (0.147 − 1.09i)28-s + (−1.95 + 0.355i)31-s + (1.53 − 0.103i)37-s + (−1.01 − 0.306i)43-s + (0.147 − 0.154i)49-s + (0.396 + 0.630i)52-s + (0.573 − 1.91i)61-s + (0.998 + 0.0448i)64-s + (0.668 + 0.217i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3789 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3789 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3789\)    =    \(3^{2} \cdot 421\)
Sign: $-0.187 + 0.982i$
Analytic conductor: \(1.89095\)
Root analytic conductor: \(1.37512\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3789} (2989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3789,\ (\ :0),\ -0.187 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3426020383\)
\(L(\frac12)\) \(\approx\) \(0.3426020383\)
\(L(1)\) 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 \( 1 + (0.919 + 0.393i)T \)
good2 \( 1 + (0.512 - 0.858i)T^{2} \)
5 \( 1 + (-0.473 + 0.880i)T^{2} \)
7 \( 1 + (1.01 - 0.433i)T + (0.691 - 0.722i)T^{2} \)
11 \( 1 + (0.134 + 0.990i)T^{2} \)
13 \( 1 + (-0.338 + 0.663i)T + (-0.587 - 0.809i)T^{2} \)
17 \( 1 + (0.995 + 0.0896i)T^{2} \)
19 \( 1 + (1.84 + 0.378i)T + (0.919 + 0.393i)T^{2} \)
23 \( 1 + (-0.178 + 0.983i)T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + (1.95 - 0.355i)T + (0.936 - 0.351i)T^{2} \)
37 \( 1 + (-1.53 + 0.103i)T + (0.990 - 0.134i)T^{2} \)
41 \( 1 + (-0.919 - 0.393i)T^{2} \)
43 \( 1 + (1.01 + 0.306i)T + (0.834 + 0.550i)T^{2} \)
47 \( 1 + (-0.880 - 0.473i)T^{2} \)
53 \( 1 + (0.722 + 0.691i)T^{2} \)
59 \( 1 + (0.657 + 0.753i)T^{2} \)
61 \( 1 + (-0.573 + 1.91i)T + (-0.834 - 0.550i)T^{2} \)
67 \( 1 + (-0.668 - 0.217i)T + (0.809 + 0.587i)T^{2} \)
71 \( 1 + (0.657 - 0.753i)T^{2} \)
73 \( 1 + (-0.309 + 1.95i)T + (-0.951 - 0.309i)T^{2} \)
79 \( 1 + (-0.252 + 0.468i)T + (-0.550 - 0.834i)T^{2} \)
83 \( 1 + (-0.657 - 0.753i)T^{2} \)
89 \( 1 + (0.351 - 0.936i)T^{2} \)
97 \( 1 + (1.09 - 1.65i)T + (-0.393 - 0.919i)T^{2} \)
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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

−8.407795178739314734237068182950, −7.981470983721591788899029659766, −6.89237214156961087496979690079, −6.39586223500581122043449016829, −5.47905786433026979632389495029, −4.55803562103837548820310172706, −3.75585856482524330382515053751, −3.04839390396533113550839602124, −2.18652750687528092831871038620, −0.19917587578544637654728454622, 1.32949267918889134408456985338, 2.40014353919306288154508489518, 3.75709528807269974931845019210, 4.15692430133443031988111476663, 5.19694386860154901946623129827, 6.00748052979752978987152264580, 6.56607586850177936866795566383, 7.22300062398262233731169677723, 8.375754277160016058283616370018, 8.955764807995124958043050145535

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