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Properties

Label 2-1456-364.191-c0-0-0
Degree $2$
Conductor $1456$
Sign $0.665 - 0.746i$
Analytic cond. $0.726638$
Root an. cond. $0.852431$
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 + (0.5 − 0.866i)5-s + (0.866 + 0.5i)7-s + i·11-s − 13-s + (0.866 + 0.5i)15-s − i·19-s + (−0.5 + 0.866i)21-s + i·27-s + (0.5 − 0.866i)29-s + (0.866 − 0.5i)31-s − 33-s + (0.866 − 0.499i)35-s − i·39-s + (−0.5 + 0.866i)41-s + ⋯
L(s)  = 1  + i·3-s + (0.5 − 0.866i)5-s + (0.866 + 0.5i)7-s + i·11-s − 13-s + (0.866 + 0.5i)15-s − i·19-s + (−0.5 + 0.866i)21-s + i·27-s + (0.5 − 0.866i)29-s + (0.866 − 0.5i)31-s − 33-s + (0.866 − 0.499i)35-s − i·39-s + (−0.5 + 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.665 - 0.746i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :0),\ 0.665 - 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.305421422\)
\(L(\frac12)\) \(\approx\) \(1.305421422\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + T \)
good3 \( 1 - iT - T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 - iT - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + iT - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + 2iT - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.812754938536737982908757114868, −9.194461803963732535989782814249, −8.414947183872296131291368117426, −7.52963338481136194911682343720, −6.50975846788356732611742515449, −5.14404531317145578844435407255, −4.91959237591574756205372304323, −4.25543626848552874352262781811, −2.69897570133037159832252470182, −1.62509413248932048358582577005, 1.29249593927034745997319876044, 2.29941484235691273198033648543, 3.35206567248361859406728073149, 4.63340198902716274886970558540, 5.63720214803177410569828149293, 6.59786379521624954075237342357, 7.06436936342267481976132625429, 7.948654140747658217529894718975, 8.484109693019835101251260054260, 9.809318891428078187683720707862

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