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Properties

Label 2-1456-364.191-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(1.003704112\)
\(L(\frac12)\) \(\approx\) \(1.003704112\)
\(L(1)\) 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

−9.492245152557022769509491193732, −8.527679750708052227168656934937, −7.76340353278750038275930689300, −6.99304025933800675982076760168, −6.19231568254702282600662536832, −5.47878326084869994675009739980, −4.34707417287146462550644442550, −3.21850968828273495971458476973, −1.97463509039108136666416439251, −0.816995846450064201946637804325, 2.24859283698279682045640784053, 2.98220701282475967688459126897, 4.09661419654583266190905604086, 4.96579604441412625217191213842, 5.80914878337072297206364943011, 7.03348519136255163784166585237, 7.13122170723425677730518735973, 8.772281760819523663476962009731, 9.455090023551358122892794338740, 9.973045502600555291853533620997

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