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Properties

Label 2-3960-440.109-c0-0-7
Degree $2$
Conductor $3960$
Sign $-i$
Analytic cond. $1.97629$
Root an. cond. $1.40580$
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·2-s − 4-s + i·5-s + 2·7-s − i·8-s − 10-s − i·11-s + 2i·14-s + 16-s − i·20-s + 22-s − 25-s − 2·28-s + 2·31-s + i·32-s + ⋯
L(s)  = 1  + i·2-s − 4-s + i·5-s + 2·7-s − i·8-s − 10-s − i·11-s + 2i·14-s + 16-s − i·20-s + 22-s − 25-s − 2·28-s + 2·31-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3960\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-i$
Analytic conductor: \(1.97629\)
Root analytic conductor: \(1.40580\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3960} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3960,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.505336606\)
\(L(\frac12)\) \(\approx\) \(1.505336606\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.332597739475181929107133254285, −8.164318686580281447868451982385, −7.43773208708229555394699808320, −6.62835101086678887862020391294, −5.93729789456770226597164268036, −5.17147362267924256959092263516, −4.48338386408034951493369863720, −3.61688609705468385581572613931, −2.50981962471622191764769163691, −1.19201641050947408877953704260, 1.15558673807892511001808746899, 1.76413640980303348561429978669, 2.67413679028139330027831027805, 4.21384490743126231566117374288, 4.50394828099472526858382166202, 5.09413490146369778547891403753, 5.87011552526503377948278442886, 7.36518257735511160934061330984, 7.961816500586970972920250543258, 8.580414721650511879668141091532

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