Properties

Label 2-4032-48.11-c1-0-25
Degree $2$
Conductor $4032$
Sign $0.936 - 0.350i$
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  + (0.871 − 0.871i)5-s + 7-s + (0.987 + 0.987i)11-s + (0.526 − 0.526i)13-s + 5.93i·17-s + (−2.78 − 2.78i)19-s − 8.59i·23-s + 3.48i·25-s + (5.07 + 5.07i)29-s + 7.72i·31-s + (0.871 − 0.871i)35-s + (3.36 + 3.36i)37-s + 4.55·41-s + (−1.26 + 1.26i)43-s + 5.02·47-s + ⋯
L(s)  = 1  + (0.389 − 0.389i)5-s + 0.377·7-s + (0.297 + 0.297i)11-s + (0.146 − 0.146i)13-s + 1.43i·17-s + (−0.638 − 0.638i)19-s − 1.79i·23-s + 0.696i·25-s + (0.942 + 0.942i)29-s + 1.38i·31-s + (0.147 − 0.147i)35-s + (0.553 + 0.553i)37-s + 0.711·41-s + (−0.193 + 0.193i)43-s + 0.733·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.936 - 0.350i$
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: \((2,\ 4032,\ (\ :1/2),\ 0.936 - 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.184036923\)
\(L(\frac12)\) \(\approx\) \(2.184036923\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.871 + 0.871i)T - 5iT^{2} \)
11 \( 1 + (-0.987 - 0.987i)T + 11iT^{2} \)
13 \( 1 + (-0.526 + 0.526i)T - 13iT^{2} \)
17 \( 1 - 5.93iT - 17T^{2} \)
19 \( 1 + (2.78 + 2.78i)T + 19iT^{2} \)
23 \( 1 + 8.59iT - 23T^{2} \)
29 \( 1 + (-5.07 - 5.07i)T + 29iT^{2} \)
31 \( 1 - 7.72iT - 31T^{2} \)
37 \( 1 + (-3.36 - 3.36i)T + 37iT^{2} \)
41 \( 1 - 4.55T + 41T^{2} \)
43 \( 1 + (1.26 - 1.26i)T - 43iT^{2} \)
47 \( 1 - 5.02T + 47T^{2} \)
53 \( 1 + (2.07 - 2.07i)T - 53iT^{2} \)
59 \( 1 + (6.52 + 6.52i)T + 59iT^{2} \)
61 \( 1 + (-8.74 + 8.74i)T - 61iT^{2} \)
67 \( 1 + (-6.20 - 6.20i)T + 67iT^{2} \)
71 \( 1 + 1.72iT - 71T^{2} \)
73 \( 1 - 10.5iT - 73T^{2} \)
79 \( 1 - 5.03iT - 79T^{2} \)
83 \( 1 + (-10.2 + 10.2i)T - 83iT^{2} \)
89 \( 1 - 2.97T + 89T^{2} \)
97 \( 1 + 8.71T + 97T^{2} \)
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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

−8.532665105605052359708283948815, −7.964286043151830263968762875216, −6.75584889017869355167306965620, −6.46998847986390670528120186059, −5.42602339643805011903151123275, −4.72587397588588935770949399066, −4.04584362953504908267948932329, −2.93339675626955966958434335106, −1.93226372494471185924575314491, −1.02203397350620911878579710576, 0.74876727138663270283336031696, 2.02027537563284461840965046881, 2.77659592824631046790716210384, 3.85580203457572009349351907410, 4.55507362900217326679799405644, 5.61632193230427416115264048475, 6.07764027226863152781832496501, 6.96173350655747654080671559486, 7.69662242875976729767497038906, 8.287484233364337911196015321640

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