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Properties

Label 2-3840-15.14-c0-0-3
Degree $2$
Conductor $3840$
Sign $1$
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  − i·3-s + i·5-s − 9-s + 15-s + 2·23-s − 25-s + i·27-s + 2i·29-s − 2i·43-s − i·45-s + 2·47-s + 49-s + 2i·67-s − 2i·69-s + i·75-s + ⋯
L(s)  = 1  − i·3-s + i·5-s − 9-s + 15-s + 2·23-s − 25-s + i·27-s + 2i·29-s − 2i·43-s − i·45-s + 2·47-s + 49-s + 2i·67-s − 2i·69-s + i·75-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $1$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (3329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.246669176\)
\(L(\frac12)\) \(\approx\) \(1.246669176\)
\(L(1)\) 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 \( 1 + iT \)
5 \( 1 - iT \)
good7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - 2T + T^{2} \)
29 \( 1 - 2iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 2iT - T^{2} \)
47 \( 1 - 2T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 2iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
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

−8.815605378150077745176798134046, −7.65942888837444445613351455006, −7.01754204371703465944853018573, −6.84532313100644106640949245329, −5.73939792958227502205846794716, −5.19438549741416169212925607599, −3.82739626350198205000857293738, −2.99518096199693647980945657087, −2.31274370408054287204785803952, −1.12127492659843845001839173917, 0.861930269953059335073922232095, 2.35909283230605317562624558027, 3.31673430582043928500690988409, 4.32319360571361173591249317505, 4.71629996617010633646601937752, 5.58571706957251820920559843344, 6.17419229198896912771458489542, 7.35225521314762163027192921704, 8.131882253346117874243983537085, 8.803172252870795808587087045302

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