Properties

Label 2-4032-48.35-c1-0-24
Degree $2$
Conductor $4032$
Sign $0.910 - 0.414i$
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 + (1.89 − 1.89i)11-s + (0.853 + 0.853i)13-s + 2.59i·17-s + 0.206i·23-s − 0.999i·25-s + (3.10 − 3.10i)29-s − 1.70i·31-s + (1.41 + 1.41i)35-s + (−1 + i)37-s + 11.0·41-s + (7.12 + 7.12i)43-s − 3.24·47-s + 49-s + ⋯
L(s)  = 1  + (0.632 + 0.632i)5-s + 0.377·7-s + (0.572 − 0.572i)11-s + (0.236 + 0.236i)13-s + 0.628i·17-s + 0.0431i·23-s − 0.199i·25-s + (0.576 − 0.576i)29-s − 0.306i·31-s + (0.239 + 0.239i)35-s + (−0.164 + 0.164i)37-s + 1.72·41-s + (1.08 + 1.08i)43-s − 0.472·47-s + 0.142·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.910 - 0.414i$
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: \((2,\ 4032,\ (\ :1/2),\ 0.910 - 0.414i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.488349143\)
\(L(\frac12)\) \(\approx\) \(2.488349143\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.89 + 1.89i)T - 11iT^{2} \)
13 \( 1 + (-0.853 - 0.853i)T + 13iT^{2} \)
17 \( 1 - 2.59iT - 17T^{2} \)
19 \( 1 - 19iT^{2} \)
23 \( 1 - 0.206iT - 23T^{2} \)
29 \( 1 + (-3.10 + 3.10i)T - 29iT^{2} \)
31 \( 1 + 1.70iT - 31T^{2} \)
37 \( 1 + (1 - i)T - 37iT^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 + (-7.12 - 7.12i)T + 43iT^{2} \)
47 \( 1 + 3.24T + 47T^{2} \)
53 \( 1 + (-0.722 - 0.722i)T + 53iT^{2} \)
59 \( 1 + (-5.00 + 5.00i)T - 59iT^{2} \)
61 \( 1 + (3.14 + 3.14i)T + 61iT^{2} \)
67 \( 1 + (4.14 - 4.14i)T - 67iT^{2} \)
71 \( 1 - 10.0iT - 71T^{2} \)
73 \( 1 + 5.66iT - 73T^{2} \)
79 \( 1 - 3.41iT - 79T^{2} \)
83 \( 1 + (7.27 + 7.27i)T + 83iT^{2} \)
89 \( 1 - 17.1T + 89T^{2} \)
97 \( 1 + 16.5T + 97T^{2} \)
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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

−8.463682649567083483775284850469, −7.82116215086521825208645198123, −6.92468358243631604364874131536, −6.12809432745204108284327654430, −5.86017600983902621581993800679, −4.63500624147138295523503024363, −3.93058214996612992589164343195, −2.92169242564145315767383705737, −2.09135353004027991184080017082, −1.00671678273470502183618657189, 0.905320382669864793567325801441, 1.79094298529089141311874978066, 2.77587813683725636525739466410, 3.91786644519296918786672203224, 4.70269458287180833492725589719, 5.38389695818395172318156790849, 6.07718452255871315085625614743, 7.00456088387680390415260936601, 7.60164872023816403320219904352, 8.550104013135351862729349596287

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