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Properties

Label 2-4032-48.35-c1-0-23
Degree $2$
Conductor $4032$
Sign $0.618 - 0.785i$
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  + (2.53 + 2.53i)5-s + 7-s + (0.490 − 0.490i)11-s + (−4.21 − 4.21i)13-s + 6.71i·17-s + (5.38 − 5.38i)19-s − 1.37i·23-s + 7.88i·25-s + (−1.45 + 1.45i)29-s + 2.66i·31-s + (2.53 + 2.53i)35-s + (2.41 − 2.41i)37-s − 1.51·41-s + (3.40 + 3.40i)43-s + 13.2·47-s + ⋯
L(s)  = 1  + (1.13 + 1.13i)5-s + 0.377·7-s + (0.148 − 0.148i)11-s + (−1.16 − 1.16i)13-s + 1.62i·17-s + (1.23 − 1.23i)19-s − 0.285i·23-s + 1.57i·25-s + (−0.270 + 0.270i)29-s + 0.477i·31-s + (0.429 + 0.429i)35-s + (0.397 − 0.397i)37-s − 0.236·41-s + (0.519 + 0.519i)43-s + 1.93·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.618 - 0.785i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (1583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ 0.618 - 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.500349717\)
\(L(\frac12)\) \(\approx\) \(2.500349717\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good5 \( 1 + (-2.53 - 2.53i)T + 5iT^{2} \)
11 \( 1 + (-0.490 + 0.490i)T - 11iT^{2} \)
13 \( 1 + (4.21 + 4.21i)T + 13iT^{2} \)
17 \( 1 - 6.71iT - 17T^{2} \)
19 \( 1 + (-5.38 + 5.38i)T - 19iT^{2} \)
23 \( 1 + 1.37iT - 23T^{2} \)
29 \( 1 + (1.45 - 1.45i)T - 29iT^{2} \)
31 \( 1 - 2.66iT - 31T^{2} \)
37 \( 1 + (-2.41 + 2.41i)T - 37iT^{2} \)
41 \( 1 + 1.51T + 41T^{2} \)
43 \( 1 + (-3.40 - 3.40i)T + 43iT^{2} \)
47 \( 1 - 13.2T + 47T^{2} \)
53 \( 1 + (-9.42 - 9.42i)T + 53iT^{2} \)
59 \( 1 + (5.46 - 5.46i)T - 59iT^{2} \)
61 \( 1 + (-8.16 - 8.16i)T + 61iT^{2} \)
67 \( 1 + (-2.47 + 2.47i)T - 67iT^{2} \)
71 \( 1 + 9.71iT - 71T^{2} \)
73 \( 1 + 2.38iT - 73T^{2} \)
79 \( 1 - 4.70iT - 79T^{2} \)
83 \( 1 + (-7.75 - 7.75i)T + 83iT^{2} \)
89 \( 1 + 5.36T + 89T^{2} \)
97 \( 1 + 3.44T + 97T^{2} \)
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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

−8.620414943530078773623459046784, −7.52735471310722522995581841948, −7.23555260628145907126692688814, −6.23507968721330557248692536337, −5.66851670263844840031236707722, −5.01270691942327810568853079124, −3.84696609121695875480795441037, −2.79332041427994135057023640515, −2.38583559771903002728119081528, −1.10468437907499560350063203198, 0.807629942610108696873644938129, 1.83800343062575048771105491284, 2.50065433096402964113607231335, 3.89712019488819149577251486266, 4.79577948835362758418810247396, 5.27818881638208804736473343089, 5.89297789979808880598549325496, 7.01431113225545085405123220140, 7.50775342620661620435372881818, 8.474372616685763831598596340050

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