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Properties

Label 2-3960-1320.1019-c0-0-4
Degree $2$
Conductor $3960$
Sign $0.994 - 0.100i$
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  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.587 + 0.809i)5-s + (1.69 − 0.550i)7-s + (0.309 − 0.951i)8-s − i·10-s + (0.453 − 0.891i)11-s + (−1.16 + 1.59i)13-s + (−1.69 − 0.550i)14-s + (−0.809 + 0.587i)16-s + (1.53 + 0.5i)19-s + (−0.587 + 0.809i)20-s + (−0.891 + 0.453i)22-s + 0.618i·23-s + (−0.309 + 0.951i)25-s + (1.87 − 0.610i)26-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.587 + 0.809i)5-s + (1.69 − 0.550i)7-s + (0.309 − 0.951i)8-s − i·10-s + (0.453 − 0.891i)11-s + (−1.16 + 1.59i)13-s + (−1.69 − 0.550i)14-s + (−0.809 + 0.587i)16-s + (1.53 + 0.5i)19-s + (−0.587 + 0.809i)20-s + (−0.891 + 0.453i)22-s + 0.618i·23-s + (−0.309 + 0.951i)25-s + (1.87 − 0.610i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3960\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.994 - 0.100i$
Analytic conductor: \(1.97629\)
Root analytic conductor: \(1.40580\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3960} (2339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3960,\ (\ :0),\ 0.994 - 0.100i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.222487213\)
\(L(\frac12)\) \(\approx\) \(1.222487213\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 \)
5 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 + (-0.453 + 0.891i)T \)
good7 \( 1 + (-1.69 + 0.550i)T + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (1.16 - 1.59i)T + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
23 \( 1 - 0.618iT - T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.0966 + 0.297i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.280 + 0.863i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.297 - 0.0966i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + 0.907iT - T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−8.801543038162235391987793566959, −7.81230172566757876954732463283, −7.39922059789527228806785959730, −6.78298247097957042208455760970, −5.71442821493193925798521400871, −4.75274685756813699226335606337, −3.90688002200632238424936566068, −2.99788156304597302182762857272, −1.91094678613372930944682346103, −1.39260354073603368444943570862, 1.03740206028114100616904381345, 1.87316727869945115540589327442, 2.74505319230986485198041354632, 4.68283745086015150727326450860, 5.03013896388662230735974985692, 5.46240523058045060012687780719, 6.48318354553104386425696098815, 7.49193627508750425941811139937, 7.953666272693591846515477156526, 8.450656966905142275105647565388

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