Properties

Label 2-4032-168.125-c1-0-4
Degree $2$
Conductor $4032$
Sign $-0.923 - 0.383i$
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  + 2i·5-s + (−2.23 − 1.41i)7-s + 1.41·11-s + 4.47·17-s − 6.32·19-s − 3.16i·23-s + 25-s − 3.16·29-s + 5.65i·31-s + (2.82 − 4.47i)35-s + 4.47i·37-s + 4.47·41-s + 4i·43-s − 8·47-s + (3.00 + 6.32i)49-s + ⋯
L(s)  = 1  + 0.894i·5-s + (−0.845 − 0.534i)7-s + 0.426·11-s + 1.08·17-s − 1.45·19-s − 0.659i·23-s + 0.200·25-s − 0.587·29-s + 1.01i·31-s + (0.478 − 0.755i)35-s + 0.735i·37-s + 0.698·41-s + 0.609i·43-s − 1.16·47-s + (0.428 + 0.903i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-0.923 - 0.383i$
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: \((2,\ 4032,\ (\ :1/2),\ -0.923 - 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5809510763\)
\(L(\frac12)\) \(\approx\) \(0.5809510763\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 1.41i)T \)
good5 \( 1 - 2iT - 5T^{2} \)
11 \( 1 - 1.41T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 4.47T + 17T^{2} \)
19 \( 1 + 6.32T + 19T^{2} \)
23 \( 1 + 3.16iT - 23T^{2} \)
29 \( 1 + 3.16T + 29T^{2} \)
31 \( 1 - 5.65iT - 31T^{2} \)
37 \( 1 - 4.47iT - 37T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 9.48T + 53T^{2} \)
59 \( 1 - 8.94iT - 59T^{2} \)
61 \( 1 - 8.48T + 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 + 9.48iT - 71T^{2} \)
73 \( 1 + 6.32iT - 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 - 6.32iT - 97T^{2} \)
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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

−8.765329869925217352808779270804, −7.947869081078980635840721743530, −7.17997395579988884067690051026, −6.48257738723426038013565309369, −6.17743146352391405211513534180, −4.97145137127583687308874625849, −4.05204350103147380394564060876, −3.30904915505275517370560747893, −2.62883721522547116458857448311, −1.31456706267932608709816729092, 0.17012476403075167705947469838, 1.45720189999462311747470159930, 2.50650010967701048603588579740, 3.56967404503176473635842882283, 4.23810159466417494557910090388, 5.26443588056846292049763839751, 5.83535662256393917912039323671, 6.56265777065597142507143842233, 7.41460367529436297751037446363, 8.320725009478967438840234419146

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