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Properties

Label 1-1456-1456.1413-r1-0-0
Degree $1$
Conductor $1456$
Sign $-0.990 + 0.136i$
Analytic cond. $156.468$
Root an. cond. $156.468$
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  + (0.866 − 0.5i)3-s − i·5-s + (0.5 − 0.866i)9-s + (0.866 − 0.5i)11-s + (−0.5 − 0.866i)15-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (0.5 + 0.866i)23-s − 25-s − i·27-s + (−0.866 + 0.5i)29-s − 31-s + (0.5 − 0.866i)33-s + (0.866 − 0.5i)37-s + (−0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s − i·5-s + (0.5 − 0.866i)9-s + (0.866 − 0.5i)11-s + (−0.5 − 0.866i)15-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (0.5 + 0.866i)23-s − 25-s − i·27-s + (−0.866 + 0.5i)29-s − 31-s + (0.5 − 0.866i)33-s + (0.866 − 0.5i)37-s + (−0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.990 + 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.990 + 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $-0.990 + 0.136i$
Analytic conductor: \(156.468\)
Root analytic conductor: \(156.468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (1413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1456,\ (1:\ ),\ -0.990 + 0.136i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1483246391 - 2.157015009i\)
\(L(\frac12)\) \(\approx\) \(-0.1483246391 - 2.157015009i\)
\(L(1)\) \(\approx\) \(1.157631512 - 0.7194931680i\)
\(L(1)\) \(\approx\) \(1.157631512 - 0.7194931680i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−20.97721522402354840715949438190, −20.058046554683711809665967447150, −19.44059670576090294911969893754, −18.794787531349718253388955018185, −18.1032788836556680386926365656, −16.93334764504097377188833089337, −16.49258033010356040181664296291, −15.21177044815243658928322313072, −14.71132466590607576753268388421, −14.54972051183556802627159072422, −13.36712386604567340648718492939, −12.69533226591620521880629019198, −11.553442495659049660792402976441, −10.783859565073344868601427363094, −10.03145917984843204148931890482, −9.46228498419750150018904091760, −8.420065261562839629426538416861, −7.78472488557193926496875433063, −6.80398002787290713525312641483, −6.14156808508113874844600368378, −4.84158933757098124043025065337, −3.90107574666857613932631317958, −3.3811880771892312458758728727, −2.30028246492567338147052335526, −1.57933765592579578333043548819, 0.33257291373379794140348265002, 1.26983212365185908293533341693, 2.04678200006460572632436044481, 3.27287938909656758485063235983, 3.94728627301736265186838998038, 4.99696185851436640883288930823, 5.92020939867907196585628054018, 6.96129920640657127459523001399, 7.652052945620822587282460310518, 8.60986649053918945843152369804, 9.12813776264310465914263764779, 9.6721250167800590309339988165, 11.04445178636172542979264406009, 11.87184148437919797482444042540, 12.60890338826663472963747263009, 13.31266492317533651160138414321, 13.90914264780212195468322933798, 14.77492858088228809159041860468, 15.4670085661129865533893555915, 16.503332327601400876790335081039, 16.97733180612112484257339178100, 17.97950551222223313142997648833, 18.72628633056143421664460530567, 19.62379060051064667447528702097, 19.91599277038182614044603693029

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