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Properties

Label 2-3072-24.5-c0-0-3
Degree $2$
Conductor $3072$
Sign $i$
Analytic cond. $1.53312$
Root an. cond. $1.23819$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s + 1.41·7-s − 9-s − 1.41i·13-s − 1.41i·21-s − 25-s + i·27-s + 1.41·31-s − 1.41i·37-s − 1.41·39-s + 1.00·49-s − 1.41i·61-s − 1.41·63-s + 2i·67-s + i·75-s + ⋯
L(s)  = 1  − i·3-s + 1.41·7-s − 9-s − 1.41i·13-s − 1.41i·21-s − 25-s + i·27-s + 1.41·31-s − 1.41i·37-s − 1.41·39-s + 1.00·49-s − 1.41i·61-s − 1.41·63-s + 2i·67-s + i·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3072} (2561, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.371008418\)
\(L(\frac12)\) \(\approx\) \(1.371008418\)
\(L(1)\) 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 \( 1 + iT \)
good5 \( 1 + T^{2} \)
7 \( 1 - 1.41T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 1.41T + T^{2} \)
37 \( 1 + 1.41iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + 1.41iT - T^{2} \)
67 \( 1 - 2iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.307428805595260966811608517978, −7.995739535510183729088866536210, −7.43534325806091729133520560719, −6.45875532961578389444087340498, −5.60639985760318732176264640059, −5.09610068188011642655209069574, −3.97249089217393240417682881293, −2.80740529283516045143920106308, −1.96262604315949143476978516717, −0.906427848062589209377445605044, 1.55699418510688602566762805720, 2.56914851219594839747133108948, 3.77997225217468893239435086849, 4.53825694721543134505339087014, 4.94132848580385952814047749961, 5.94058279434192753567123738636, 6.74400794974121106263974813836, 7.87429824831286000687818514938, 8.328548130650305971677988150378, 9.118348081012551002328896016643

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