Properties

Label 2-4032-21.20-c1-0-30
Degree $2$
Conductor $4032$
Sign $0.537 - 0.843i$
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  + 3.46·5-s + (1 + 2.44i)7-s − 4.24i·11-s + 4.89i·13-s − 3.46·17-s + 4.89i·19-s − 4.24i·23-s + 6.99·25-s + 4.24i·29-s + (3.46 + 8.48i)35-s + 8·37-s + 3.46·41-s − 2·43-s − 6.92·47-s + (−4.99 + 4.89i)49-s + ⋯
L(s)  = 1  + 1.54·5-s + (0.377 + 0.925i)7-s − 1.27i·11-s + 1.35i·13-s − 0.840·17-s + 1.12i·19-s − 0.884i·23-s + 1.39·25-s + 0.787i·29-s + (0.585 + 1.43i)35-s + 1.31·37-s + 0.541·41-s − 0.304·43-s − 1.01·47-s + (−0.714 + 0.699i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.537 - 0.843i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (3905, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ 0.537 - 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.623834852\)
\(L(\frac12)\) \(\approx\) \(2.623834852\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 2.44i)T \)
good5 \( 1 - 3.46T + 5T^{2} \)
11 \( 1 + 4.24iT - 11T^{2} \)
13 \( 1 - 4.89iT - 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 - 4.89iT - 19T^{2} \)
23 \( 1 + 4.24iT - 23T^{2} \)
29 \( 1 - 4.24iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 - 12.7iT - 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 - 9.79iT - 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 4.24iT - 71T^{2} \)
73 \( 1 - 4.89iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 6.92T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + 4.89iT - 97T^{2} \)
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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

−8.791874088316280320529768067205, −8.049913272803025881435177147627, −6.80538604573055589154557134920, −6.18244005409163404862664763096, −5.78506192910887284201842792161, −4.97025084353375584785565688759, −4.05537498601006939298680794673, −2.78885286239999672440940975005, −2.15089762282118133501820677163, −1.30140248096369415620708328802, 0.75311999981005301081181906987, 1.91168199655078263376519365729, 2.52986387098497523754355384376, 3.73394319421718863815917825658, 4.82545676169597095628216853129, 5.16814440814091050452836183426, 6.19202716123330482541095835035, 6.81123471744281439282065153410, 7.54048995694271467134440822766, 8.261177192869825332621064490666

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