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Properties

Label 2-3840-120.29-c0-0-7
Degree $2$
Conductor $3840$
Sign $i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)3-s + 5-s − 1.41i·7-s − 1.00i·9-s + (0.707 − 0.707i)15-s + (−1.00 − 1.00i)21-s − 1.41·23-s + 25-s + (−0.707 − 0.707i)27-s − 1.41i·35-s + 2i·41-s + 1.41·43-s − 1.00i·45-s − 1.41·47-s − 1.00·49-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + 5-s − 1.41i·7-s − 1.00i·9-s + (0.707 − 0.707i)15-s + (−1.00 − 1.00i)21-s − 1.41·23-s + 25-s + (−0.707 − 0.707i)27-s − 1.41i·35-s + 2i·41-s + 1.41·43-s − 1.00i·45-s − 1.41·47-s − 1.00·49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (1409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.921168542\)
\(L(\frac12)\) \(\approx\) \(1.921168542\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 - T \)
good7 \( 1 + 1.41iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.41T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 2iT - T^{2} \)
43 \( 1 - 1.41T + T^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.224138613059314120535425253187, −7.87793391882344343218197106631, −6.93298193775917072010881596003, −6.49012643162770453133178354126, −5.69806722715290861223027430679, −4.55115631828780238188888458981, −3.78267561225530560880067775691, −2.84657658606152548880546688884, −1.89070429152671077277125376637, −1.02085679069247313545941175519, 1.92457772296514710647893811046, 2.36467468243838581399865719853, 3.28737238972921153090062467326, 4.27993928293286249973162564634, 5.25997874085027391760564002322, 5.69111673677878598367374102199, 6.45948655295759933564185341859, 7.56369922480742023468637185824, 8.395593147872245024782895728816, 8.921025948570608405527327784822

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