↑

Properties

Label 2-3840-3840.2309-c0-0-0
Degree $2$
Conductor $3840$
Sign $0.985 + 0.170i$
Analytic cond. $1.91640$
Root an. cond. $1.38434$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.903 + 0.427i)2-s + (0.146 − 0.989i)3-s + (0.634 − 0.773i)4-s + (−0.998 − 0.0490i)5-s + (0.290 + 0.956i)6-s + (−0.242 + 0.970i)8-s + (−0.956 − 0.290i)9-s + (0.923 − 0.382i)10-s + (−0.671 − 0.740i)12-s + (−0.195 + 0.980i)15-s + (−0.195 − 0.980i)16-s + (0.289 + 1.45i)17-s + (0.989 − 0.146i)18-s + (0.207 + 0.439i)19-s + (−0.671 + 0.740i)20-s + ⋯
L(s)  = 1  + (−0.903 + 0.427i)2-s + (0.146 − 0.989i)3-s + (0.634 − 0.773i)4-s + (−0.998 − 0.0490i)5-s + (0.290 + 0.956i)6-s + (−0.242 + 0.970i)8-s + (−0.956 − 0.290i)9-s + (0.923 − 0.382i)10-s + (−0.671 − 0.740i)12-s + (−0.195 + 0.980i)15-s + (−0.195 − 0.980i)16-s + (0.289 + 1.45i)17-s + (0.989 − 0.146i)18-s + (0.207 + 0.439i)19-s + (−0.671 + 0.740i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $0.985 + 0.170i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (2309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :0),\ 0.985 + 0.170i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6604217150\)
\(L(\frac12)\) \(\approx\) \(0.6604217150\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.903 - 0.427i)T \)
3 \( 1 + (-0.146 + 0.989i)T \)
5 \( 1 + (0.998 + 0.0490i)T \)
good7 \( 1 + (-0.555 - 0.831i)T^{2} \)
11 \( 1 + (0.471 - 0.881i)T^{2} \)
13 \( 1 + (0.0980 + 0.995i)T^{2} \)
17 \( 1 + (-0.289 - 1.45i)T + (-0.923 + 0.382i)T^{2} \)
19 \( 1 + (-0.207 - 0.439i)T + (-0.634 + 0.773i)T^{2} \)
23 \( 1 + (0.520 + 0.427i)T + (0.195 + 0.980i)T^{2} \)
29 \( 1 + (-0.881 + 0.471i)T^{2} \)
31 \( 1 + (-1.42 - 0.591i)T + (0.707 + 0.707i)T^{2} \)
37 \( 1 + (0.773 - 0.634i)T^{2} \)
41 \( 1 + (-0.980 + 0.195i)T^{2} \)
43 \( 1 + (-0.956 + 0.290i)T^{2} \)
47 \( 1 + (-0.661 + 0.990i)T + (-0.382 - 0.923i)T^{2} \)
53 \( 1 + (-0.914 - 0.229i)T + (0.881 + 0.471i)T^{2} \)
59 \( 1 + (0.0980 - 0.995i)T^{2} \)
61 \( 1 + (0.612 + 0.825i)T + (-0.290 + 0.956i)T^{2} \)
67 \( 1 + (-0.290 + 0.956i)T^{2} \)
71 \( 1 + (0.831 - 0.555i)T^{2} \)
73 \( 1 + (-0.555 + 0.831i)T^{2} \)
79 \( 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2} \)
83 \( 1 + (-0.644 + 1.80i)T + (-0.773 - 0.634i)T^{2} \)
89 \( 1 + (-0.195 + 0.980i)T^{2} \)
97 \( 1 + (0.707 + 0.707i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.578295210012615520072390020371, −7.79483356378443755599594605250, −7.51543328414920415679054480214, −6.50898773956875857532147537847, −6.12011491026010590008633843430, −5.11548742285944668706151601464, −3.94666967994144050975482344636, −2.94238316861300360392725392139, −1.88044472380026461251550002533, −0.857310613840104493683932138693, 0.71555231369157625272200241333, 2.50053718941235578314838234435, 3.10070752312427775977588122548, 3.97391045286354316691352943453, 4.62985694123827851750119067810, 5.64030384039491594543648555755, 6.76795646825276008721554448076, 7.49974021882900926702930034251, 8.105286206352819157397473625660, 8.767599841024583493700305401351

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