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Properties

Label 2-3840-40.29-c1-0-81
Degree $2$
Conductor $3840$
Sign $-0.707 + 0.707i$
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 − 2.23·5-s − 2i·7-s + 9-s − 4.47i·11-s + 4.47·13-s − 2.23·15-s − 4.47i·17-s − 2i·21-s + 4i·23-s + 5.00·25-s + 27-s − 4i·29-s − 8.94·31-s − 4.47i·33-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.999·5-s − 0.755i·7-s + 0.333·9-s − 1.34i·11-s + 1.24·13-s − 0.577·15-s − 1.08i·17-s − 0.436i·21-s + 0.834i·23-s + 1.00·25-s + 0.192·27-s − 0.742i·29-s − 1.60·31-s − 0.778i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $-0.707 + 0.707i$
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: \((2,\ 3840,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.350637261\)
\(L(\frac12)\) \(\approx\) \(1.350637261\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23T \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 4.47iT - 11T^{2} \)
13 \( 1 - 4.47T + 13T^{2} \)
17 \( 1 + 4.47iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 + 8.94T + 31T^{2} \)
37 \( 1 + 4.47T + 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 - 4.47T + 53T^{2} \)
59 \( 1 + 13.4iT - 59T^{2} \)
61 \( 1 - 10iT - 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 8.94T + 71T^{2} \)
73 \( 1 + 8.94iT - 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 17.8iT - 97T^{2} \)
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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

−8.274663408725340380671660227178, −7.49578862461765811052512929660, −7.01348498124115663992952685102, −6.01982517568381180672735301769, −5.15401088835524692285341568299, −4.04833174315356003816406176933, −3.59254065723351348045175197619, −2.95008982638381777433878127519, −1.41466913032730345047227626586, −0.37781453282357217875944020303, 1.47143731584982145966939072733, 2.34405603258643996119605340703, 3.51963785368362685808255002956, 3.96088128483471747821718555123, 4.89555837090095119939382923791, 5.76598395696093103810302295476, 6.81990963085777603402646468448, 7.25610501718503652000996194800, 8.249161689125554790028891133250, 8.651172178905414124407278081886

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