Properties

Label 2-4032-48.11-c1-0-16
Degree $2$
Conductor $4032$
Sign $-0.0179 - 0.999i$
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  + (−1.41 + 1.41i)5-s + 7-s + (3.97 + 3.97i)11-s + (−1.10 + 1.10i)13-s − 6.39i·17-s + 2.97i·23-s + 0.999i·25-s + (5.53 + 5.53i)29-s − 2.20i·31-s + (−1.41 + 1.41i)35-s + (−1 − i)37-s − 2.08·41-s + (4.68 − 4.68i)43-s + 8.77·47-s + 49-s + ⋯
L(s)  = 1  + (−0.632 + 0.632i)5-s + 0.377·7-s + (1.19 + 1.19i)11-s + (−0.305 + 0.305i)13-s − 1.55i·17-s + 0.620i·23-s + 0.199i·25-s + (1.02 + 1.02i)29-s − 0.396i·31-s + (−0.239 + 0.239i)35-s + (−0.164 − 0.164i)37-s − 0.325·41-s + (0.713 − 0.713i)43-s + 1.28·47-s + 0.142·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0179 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0179 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-0.0179 - 0.999i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (3599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ -0.0179 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.668175036\)
\(L(\frac12)\) \(\approx\) \(1.668175036\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - 1.41i)T - 5iT^{2} \)
11 \( 1 + (-3.97 - 3.97i)T + 11iT^{2} \)
13 \( 1 + (1.10 - 1.10i)T - 13iT^{2} \)
17 \( 1 + 6.39iT - 17T^{2} \)
19 \( 1 + 19iT^{2} \)
23 \( 1 - 2.97iT - 23T^{2} \)
29 \( 1 + (-5.53 - 5.53i)T + 29iT^{2} \)
31 \( 1 + 2.20iT - 31T^{2} \)
37 \( 1 + (1 + i)T + 37iT^{2} \)
41 \( 1 + 2.08T + 41T^{2} \)
43 \( 1 + (-4.68 + 4.68i)T - 43iT^{2} \)
47 \( 1 - 8.77T + 47T^{2} \)
53 \( 1 + (3.83 - 3.83i)T - 53iT^{2} \)
59 \( 1 + (-9.51 - 9.51i)T + 59iT^{2} \)
61 \( 1 + (5.10 - 5.10i)T - 61iT^{2} \)
67 \( 1 + (6.10 + 6.10i)T + 67iT^{2} \)
71 \( 1 - 6.62iT - 71T^{2} \)
73 \( 1 + 7.04iT - 73T^{2} \)
79 \( 1 - 4.41iT - 79T^{2} \)
83 \( 1 + (-10.0 + 10.0i)T - 83iT^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 + 15.5T + 97T^{2} \)
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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

−8.774234775565750235177432263826, −7.54656939940252281383104616861, −7.23352287170948288195089402283, −6.74385394356697677262149839666, −5.61849561429820152118637805997, −4.70887694330357294410558934432, −4.14099986344046880489154402036, −3.20360892012347565769050749661, −2.27395290560899950717877051125, −1.14891882500326072162677785212, 0.55696138102526987282621870570, 1.48948620040732756206934720785, 2.78503671848845591791711496225, 3.88485224750827175009847623293, 4.25413319485893721326317203962, 5.24520751379263464423071931609, 6.14940151782878690319331497776, 6.61820232261092876753512328091, 7.79833631765472822044877221716, 8.396196577073959741743755854760

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