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Properties

Label 2-3840-15.14-c0-0-5
Degree $2$
Conductor $3840$
Sign $1$
Analytic cond. $1.91640$
Root an. cond. $1.38434$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 9-s − 15-s + 25-s − 27-s + 2·31-s + 45-s + 49-s − 2·53-s − 75-s − 2·79-s + 81-s + 2·83-s − 2·93-s + 2·107-s + ⋯
L(s)  = 1  − 3-s + 5-s + 9-s − 15-s + 25-s − 27-s + 2·31-s + 45-s + 49-s − 2·53-s − 75-s − 2·79-s + 81-s + 2·83-s − 2·93-s + 2·107-s + ⋯

Functional equation

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

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: yes
Arithmetic: yes
Character: $\chi_{3840} (3329, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.170584942\)
\(L(\frac12)\) \(\approx\) \(1.170584942\)
\(L(1)\) 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 \( 1 + T \)
5 \( 1 - T \)
good7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 + T^{2} \)
53 \( ( 1 + T )^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 + T )^{2} \)
83 \( ( 1 - T )^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−8.756955866766645499021262404197, −7.85108435974160185013086648709, −6.96635662582057277785423470625, −6.30982934812477082591843280667, −5.81662794753957581764697476036, −4.95790272273093413739439586113, −4.38864111141211319911419524222, −3.12560613795571978238219125556, −2.05425399675574092686423508599, −1.02468077780765774089259396945, 1.02468077780765774089259396945, 2.05425399675574092686423508599, 3.12560613795571978238219125556, 4.38864111141211319911419524222, 4.95790272273093413739439586113, 5.81662794753957581764697476036, 6.30982934812477082591843280667, 6.96635662582057277785423470625, 7.85108435974160185013086648709, 8.756955866766645499021262404197

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