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Properties

Label 2-3960-5.4-c1-0-67
Degree $2$
Conductor $3960$
Sign $-0.998 + 0.0452i$
Analytic cond. $31.6207$
Root an. cond. $5.62323$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.101 − 2.23i)5-s − 3.61i·7-s − 11-s − 0.816i·13-s − 1.81i·17-s − 0.723·19-s − 4.63i·23-s + (−4.97 + 0.451i)25-s + 0.539·29-s + 9.04·31-s + (−8.07 + 0.365i)35-s − 9.34i·37-s − 1.11·41-s + 5.63i·43-s − 2.01i·47-s + ⋯
L(s)  = 1  + (−0.0452 − 0.998i)5-s − 1.36i·7-s − 0.301·11-s − 0.226i·13-s − 0.440i·17-s − 0.165·19-s − 0.966i·23-s + (−0.995 + 0.0903i)25-s + 0.100·29-s + 1.62·31-s + (−1.36 + 0.0617i)35-s − 1.53i·37-s − 0.173·41-s + 0.859i·43-s − 0.294i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0452i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3960\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.998 + 0.0452i$
Analytic conductor: \(31.6207\)
Root analytic conductor: \(5.62323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3960} (3169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3960,\ (\ :1/2),\ -0.998 + 0.0452i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.213763651\)
\(L(\frac12)\) \(\approx\) \(1.213763651\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (0.101 + 2.23i)T \)
11 \( 1 + T \)
good7 \( 1 + 3.61iT - 7T^{2} \)
13 \( 1 + 0.816iT - 13T^{2} \)
17 \( 1 + 1.81iT - 17T^{2} \)
19 \( 1 + 0.723T + 19T^{2} \)
23 \( 1 + 4.63iT - 23T^{2} \)
29 \( 1 - 0.539T + 29T^{2} \)
31 \( 1 - 9.04T + 31T^{2} \)
37 \( 1 + 9.34iT - 37T^{2} \)
41 \( 1 + 1.11T + 41T^{2} \)
43 \( 1 - 5.63iT - 43T^{2} \)
47 \( 1 + 2.01iT - 47T^{2} \)
53 \( 1 + 1.29iT - 53T^{2} \)
59 \( 1 + 7.11T + 59T^{2} \)
61 \( 1 - 3.61T + 61T^{2} \)
67 \( 1 - 2.26iT - 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 + 11.3iT - 73T^{2} \)
79 \( 1 - 3.97T + 79T^{2} \)
83 \( 1 - 2.50iT - 83T^{2} \)
89 \( 1 + 6.19T + 89T^{2} \)
97 \( 1 - 0.0431iT - 97T^{2} \)
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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

−8.028852081979971431458356147104, −7.51715643929440416266875472542, −6.67245136088507889008406678813, −5.86867735404551697887436885753, −4.83792775144416372279798957657, −4.44098523389621416629952339175, −3.58749476333353050667874754032, −2.45585132180528955356551279209, −1.17624228403865209930388803518, −0.36979971531639552493704920819, 1.64237019882859400662756937905, 2.61716776966925762533234054920, 3.15162807953014815920162333063, 4.24031132956456808128609004035, 5.21907877624769766392970568079, 5.97033151520452688486154517765, 6.49998633559746209478241185339, 7.33710350469875058697906346562, 8.178029604053932554450966592555, 8.685870821520992763738211509154

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