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Properties

Label 2-3072-48.29-c0-0-0
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: $\chi_{3072} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :0),\ -0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05510245051\)
\(L(\frac12)\) \(\approx\) \(0.05510245051\)
\(L(1)\) 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

−9.163257027921910456656207803604, −8.758411727208298707555440543814, −7.996872824343106805251382050432, −7.04164669967364827523082012224, −6.01683977857191732637890206296, −5.34035978444520794642450399716, −4.84722733721187008065904897841, −3.93458235800483918303844249446, −3.15036208210327566599643468629, −1.66235799873325136318320328669, 0.03806645544482427640495626107, 1.46846488598592477465924560733, 2.80614181105896719540616687948, 3.82976932549048358158623530197, 4.38469755411089881379616552431, 5.55254331917084768606372644647, 6.42030867004610475216415385502, 7.12312671155237505799401723435, 7.55268907963192149735364372417, 7.971360320457353385646987773336

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