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Properties

Label 2-3789-421.263-c0-0-0
Degree $2$
Conductor $3789$
Sign $0.657 - 0.753i$
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  + (0.998 − 0.0448i)4-s + (0.691 + 1.84i)7-s + (1.79 + 0.284i)13-s + (0.995 − 0.0896i)16-s + (−0.874 + 0.605i)19-s + (−0.995 − 0.0896i)25-s + (0.773 + 1.80i)28-s + (−1.65 − 0.988i)31-s + (−0.966 − 0.197i)37-s + (1.27 − 1.52i)43-s + (−2.16 + 1.88i)49-s + (1.80 + 0.203i)52-s + (−1.50 − 1.25i)61-s + (0.990 − 0.134i)64-s + (−1.03 + 1.42i)67-s + ⋯
L(s)  = 1  + (0.998 − 0.0448i)4-s + (0.691 + 1.84i)7-s + (1.79 + 0.284i)13-s + (0.995 − 0.0896i)16-s + (−0.874 + 0.605i)19-s + (−0.995 − 0.0896i)25-s + (0.773 + 1.80i)28-s + (−1.65 − 0.988i)31-s + (−0.966 − 0.197i)37-s + (1.27 − 1.52i)43-s + (−2.16 + 1.88i)49-s + (1.80 + 0.203i)52-s + (−1.50 − 1.25i)61-s + (0.990 − 0.134i)64-s + (−1.03 + 1.42i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3789 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3789 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3789\)    =    \(3^{2} \cdot 421\)
Sign: $0.657 - 0.753i$
Analytic conductor: \(1.89095\)
Root analytic conductor: \(1.37512\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3789} (2368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3789,\ (\ :0),\ 0.657 - 0.753i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.865740108\)
\(L(\frac12)\) \(\approx\) \(1.865740108\)
\(L(1)\) 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 \( 1 + (0.351 - 0.936i)T \)
good2 \( 1 + (-0.998 + 0.0448i)T^{2} \)
5 \( 1 + (0.995 + 0.0896i)T^{2} \)
7 \( 1 + (-0.691 - 1.84i)T + (-0.753 + 0.657i)T^{2} \)
11 \( 1 + (-0.393 + 0.919i)T^{2} \)
13 \( 1 + (-1.79 - 0.284i)T + (0.951 + 0.309i)T^{2} \)
17 \( 1 + (0.963 - 0.266i)T^{2} \)
19 \( 1 + (0.874 - 0.605i)T + (0.351 - 0.936i)T^{2} \)
23 \( 1 + (0.512 + 0.858i)T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + (1.65 + 0.988i)T + (0.473 + 0.880i)T^{2} \)
37 \( 1 + (0.966 + 0.197i)T + (0.919 + 0.393i)T^{2} \)
41 \( 1 + (-0.351 + 0.936i)T^{2} \)
43 \( 1 + (-1.27 + 1.52i)T + (-0.178 - 0.983i)T^{2} \)
47 \( 1 + (-0.0896 + 0.995i)T^{2} \)
53 \( 1 + (-0.657 - 0.753i)T^{2} \)
59 \( 1 + (-0.834 - 0.550i)T^{2} \)
61 \( 1 + (1.50 + 1.25i)T + (0.178 + 0.983i)T^{2} \)
67 \( 1 + (1.03 - 1.42i)T + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.834 + 0.550i)T^{2} \)
73 \( 1 + (0.809 + 1.58i)T + (-0.587 + 0.809i)T^{2} \)
79 \( 1 + (-1.43 - 0.129i)T + (0.983 + 0.178i)T^{2} \)
83 \( 1 + (0.834 + 0.550i)T^{2} \)
89 \( 1 + (-0.880 - 0.473i)T^{2} \)
97 \( 1 + (-1.80 + 0.328i)T + (0.936 - 0.351i)T^{2} \)
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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

−8.817419458809718890624067049289, −8.075206201456092352069628504700, −7.41828027715498068878239697795, −6.18179961781687331127951883342, −6.00424251240554108703741398148, −5.34960236091105130044241214526, −4.06264940964308192675073915862, −3.24005442979596539533100254970, −1.98251089324741421453133927205, −1.84521162897248773908708074397, 1.14115747239140364117762598022, 1.83371274455399233131644642259, 3.27217556011123321298397609745, 3.84186068568047334475412944418, 4.65782003120932143831993410205, 5.79932006591163124933902375357, 6.44588681993004596083469913895, 7.20018274070416150356746058237, 7.69968339495129320886814438502, 8.382386982308874637301871890732

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