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Properties

Label 2-3072-24.5-c0-0-2
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

\(L(\frac{1}{2})\) \(\approx\) \(0.3917105309\)
\(L(\frac12)\) \(\approx\) \(0.3917105309\)
\(L(1)\) 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

−8.939769570774590924944127574301, −8.009944300642626086571270610158, −7.26604384252368887220959970768, −6.13911407563483109144561788689, −5.73177069739356318259352821008, −4.87345688908346439663757475021, −3.65967598701394398853983320033, −3.40343644802661565931282679643, −2.33788465879982988105510338708, −0.22893141895054074406452459836, 1.49574881140859379192355703465, 2.49608134150929413555285050786, 3.40100705336555851131723259615, 4.28229299873590296069668603794, 5.58017378028337630940512327483, 6.18038059550022122395350954707, 6.91231782553707221025476973272, 7.29388916794638713409114700035, 8.356801386572979281891286037583, 9.055999571818516821535578541808

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