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Properties

Label 2-4032-168.125-c1-0-56
Degree $2$
Conductor $4032$
Sign $-0.184 + 0.982i$
Analytic cond. $32.1956$
Root an. cond. $5.67412$
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  − 3.91i·5-s + (2.03 − 1.68i)7-s + 1.62·11-s + 2.54·13-s + 2.94·17-s + 2.72·19-s − 8.57i·23-s − 10.3·25-s + 8.21·29-s − 8.08i·31-s + (−6.61 − 7.98i)35-s + 8.05i·37-s + 9.17·41-s + 9.05i·43-s + 1.17·47-s + ⋯
L(s)  = 1  − 1.75i·5-s + (0.769 − 0.638i)7-s + 0.490·11-s + 0.706·13-s + 0.713·17-s + 0.624·19-s − 1.78i·23-s − 2.07·25-s + 1.52·29-s − 1.45i·31-s + (−1.11 − 1.34i)35-s + 1.32i·37-s + 1.43·41-s + 1.38i·43-s + 0.171·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-0.184 + 0.982i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ -0.184 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.517488027\)
\(L(\frac12)\) \(\approx\) \(2.517488027\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-2.03 + 1.68i)T \)
good5 \( 1 + 3.91iT - 5T^{2} \)
11 \( 1 - 1.62T + 11T^{2} \)
13 \( 1 - 2.54T + 13T^{2} \)
17 \( 1 - 2.94T + 17T^{2} \)
19 \( 1 - 2.72T + 19T^{2} \)
23 \( 1 + 8.57iT - 23T^{2} \)
29 \( 1 - 8.21T + 29T^{2} \)
31 \( 1 + 8.08iT - 31T^{2} \)
37 \( 1 - 8.05iT - 37T^{2} \)
41 \( 1 - 9.17T + 41T^{2} \)
43 \( 1 - 9.05iT - 43T^{2} \)
47 \( 1 - 1.17T + 47T^{2} \)
53 \( 1 + 2.44T + 53T^{2} \)
59 \( 1 - 1.45iT - 59T^{2} \)
61 \( 1 - 9.74T + 61T^{2} \)
67 \( 1 - 7.35iT - 67T^{2} \)
71 \( 1 - 7.71iT - 71T^{2} \)
73 \( 1 - 15.0iT - 73T^{2} \)
79 \( 1 + 0.0913T + 79T^{2} \)
83 \( 1 + 2.03iT - 83T^{2} \)
89 \( 1 - 1.48T + 89T^{2} \)
97 \( 1 + 7.53iT - 97T^{2} \)
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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

−8.232414525172994816213623574128, −7.86711024687356226922925462655, −6.72810350978827450476865912379, −5.91551074512406467419741819428, −5.10521433251818884422778366535, −4.40759264750768746576389943517, −4.01045743887502978858125244739, −2.58953926936150104049685560030, −1.14653207258234765779581038722, −0.960283641168845296236800550222, 1.33968365618327360082214805819, 2.34100107111861281842256811408, 3.26885330520431090660710563200, 3.75804036317539889188713213144, 5.05426652566982580022485143322, 5.81048616702736839632288688581, 6.41188694321808990678030015725, 7.31268955111103273097130540859, 7.67779731404461462386097257483, 8.637895570379370526750362149208

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