Properties

Label 2-3840-40.29-c1-0-25
Degree $2$
Conductor $3840$
Sign $0.948 - 0.316i$
Analytic cond. $30.6625$
Root an. cond. $5.53737$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + (−2 − i)5-s − 4i·7-s + 9-s + 4i·11-s + (−2 − i)15-s + 4i·17-s − 4i·21-s + 4i·23-s + (3 + 4i)25-s + 27-s + 6i·29-s + 4·31-s + 4i·33-s + (−4 + 8i)35-s + ⋯
L(s)  = 1  + 0.577·3-s + (−0.894 − 0.447i)5-s − 1.51i·7-s + 0.333·9-s + 1.20i·11-s + (−0.516 − 0.258i)15-s + 0.970i·17-s − 0.872i·21-s + 0.834i·23-s + (0.600 + 0.800i)25-s + 0.192·27-s + 1.11i·29-s + 0.718·31-s + 0.696i·33-s + (−0.676 + 1.35i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $0.948 - 0.316i$
Analytic conductor: \(30.6625\)
Root analytic conductor: \(5.53737\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (2689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :1/2),\ 0.948 - 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.774280956\)
\(L(\frac12)\) \(\approx\) \(1.774280956\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + (2 + i)T \)
good7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 8iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−8.438889038947173182413323343330, −7.68141460742279706607448359768, −7.25741095001970878006871203103, −6.67788901736851606326597663012, −5.30929832302428030216489175673, −4.47135175044073417687056955176, −3.94434004355725885132051536689, −3.31941608057261962450531382365, −1.89069860782359118672668043211, −0.955018373414104480512377092868, 0.59737789044940422032016358078, 2.34943906245832997555044402730, 2.83003538722097492246331937631, 3.63925703361201978622807868801, 4.59340973585300201209166650320, 5.51051806295697786845614915780, 6.26345118242283070785353508383, 7.02004650485077925950007305841, 7.965815070039479399631035195020, 8.426220001072337156911335852935

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