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Properties

Label 2-3072-96.77-c0-0-7
Degree $2$
Conductor $3072$
Sign $-0.980 + 0.195i$
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  + (0.382 − 0.923i)3-s + (−0.541 + 0.541i)7-s + (−0.707 − 0.707i)9-s + (−1.70 − 0.707i)13-s + (−1.30 − 0.541i)19-s + (0.292 + 0.707i)21-s + (0.707 − 0.707i)25-s + (−0.923 + 0.382i)27-s − 1.84·31-s + (0.707 − 0.292i)37-s + (−1.30 + 1.30i)39-s + (−0.541 − 1.30i)43-s + 0.414i·49-s + (−1 + 0.999i)57-s + (0.707 − 1.70i)61-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)3-s + (−0.541 + 0.541i)7-s + (−0.707 − 0.707i)9-s + (−1.70 − 0.707i)13-s + (−1.30 − 0.541i)19-s + (0.292 + 0.707i)21-s + (0.707 − 0.707i)25-s + (−0.923 + 0.382i)27-s − 1.84·31-s + (0.707 − 0.292i)37-s + (−1.30 + 1.30i)39-s + (−0.541 − 1.30i)43-s + 0.414i·49-s + (−1 + 0.999i)57-s + (0.707 − 1.70i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $-0.980 + 0.195i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3072} (2177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :0),\ -0.980 + 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5749064185\)
\(L(\frac12)\) \(\approx\) \(0.5749064185\)
\(L(1)\) 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 \( 1 + (-0.382 + 0.923i)T \)
good5 \( 1 + (-0.707 + 0.707i)T^{2} \)
7 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
11 \( 1 + (0.707 - 0.707i)T^{2} \)
13 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (0.707 + 0.707i)T^{2} \)
31 \( 1 + 1.84T + T^{2} \)
37 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 + 0.707i)T^{2} \)
61 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
67 \( 1 + (-0.707 - 0.707i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 - 1.84iT - T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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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

−8.563091329292372828651095760181, −7.69572402735395454050531702610, −7.09564328897137325987002109675, −6.41066772236168330503256513998, −5.59818667341485356996233756566, −4.76666611465582467856175282258, −3.54051081337334602953822783821, −2.59905501134468616708970403416, −2.08302681517729616018412495862, −0.29881124392183873162119947399, 1.96220385938437441028412551569, 2.89443320587574722603176103828, 3.82177760739180250867609845436, 4.50325189105528265974433581153, 5.21271578143901256421103587008, 6.21039353254847842582231880726, 7.12835915638628876832860189988, 7.68733743617976816166940037205, 8.735473060760002288699721772840, 9.242544750885956720214008620687

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