Properties

Label 2-3072-48.5-c0-0-6
Degree $2$
Conductor $3072$
Sign $0.923 + 0.382i$
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  + (−0.707 − 0.707i)3-s + (1 − i)5-s + 1.41i·7-s + 1.00i·9-s − 1.41·15-s + (1.00 − 1.00i)21-s i·25-s + (0.707 − 0.707i)27-s + (1 + i)29-s + 1.41·31-s + (1.41 + 1.41i)35-s + (1.00 + 1.00i)45-s − 1.00·49-s + (1 − i)53-s + (−1.41 + 1.41i)59-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + (1 − i)5-s + 1.41i·7-s + 1.00i·9-s − 1.41·15-s + (1.00 − 1.00i)21-s i·25-s + (0.707 − 0.707i)27-s + (1 + i)29-s + 1.41·31-s + (1.41 + 1.41i)35-s + (1.00 + 1.00i)45-s − 1.00·49-s + (1 − i)53-s + (−1.41 + 1.41i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3072} (1793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :0),\ 0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.202260690\)
\(L(\frac12)\) \(\approx\) \(1.202260690\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T \)
good5 \( 1 + (-1 + i)T - iT^{2} \)
7 \( 1 - 1.41iT - T^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-1 - i)T + iT^{2} \)
31 \( 1 - 1.41T + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-1 + i)T - iT^{2} \)
59 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−8.611335888380064839371072793067, −8.424014477339438577791325966892, −7.22589046270852196441784660626, −6.30316039865065811692633334538, −5.83972136251799428852648814744, −5.16052111905534477347100105754, −4.61715759243337261069415793406, −2.86688605358286930455247893522, −2.05239348523040099847752966227, −1.15830482563995193885761785435, 1.01078294170993759154394353789, 2.49897956678911766965327139983, 3.46369972840480522571364315768, 4.30521829959841082469608134852, 5.03078800469277393504516950514, 6.17259652628470540877831807922, 6.42957412965856392556412431365, 7.22421646684129117684433668203, 8.105095250311947688323747033677, 9.275637733889537795856459108930

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