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Properties

Label 1-712-712.557-r1-0-0
Degree $1$
Conductor $712$
Sign $0.929 + 0.368i$
Analytic cond. $76.5150$
Root an. cond. $76.5150$
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  + (0.599 + 0.800i)3-s + (0.540 − 0.841i)5-s + (−0.977 + 0.212i)7-s + (−0.281 + 0.959i)9-s + (0.841 − 0.540i)11-s + (−0.800 + 0.599i)13-s + (0.997 − 0.0713i)15-s + (0.755 − 0.654i)17-s + (0.479 + 0.877i)19-s + (−0.755 − 0.654i)21-s + (0.877 − 0.479i)23-s + (−0.415 − 0.909i)25-s + (−0.936 + 0.349i)27-s + (−0.212 − 0.977i)29-s + (0.877 + 0.479i)31-s + ⋯
L(s)  = 1  + (0.599 + 0.800i)3-s + (0.540 − 0.841i)5-s + (−0.977 + 0.212i)7-s + (−0.281 + 0.959i)9-s + (0.841 − 0.540i)11-s + (−0.800 + 0.599i)13-s + (0.997 − 0.0713i)15-s + (0.755 − 0.654i)17-s + (0.479 + 0.877i)19-s + (−0.755 − 0.654i)21-s + (0.877 − 0.479i)23-s + (−0.415 − 0.909i)25-s + (−0.936 + 0.349i)27-s + (−0.212 − 0.977i)29-s + (0.877 + 0.479i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.929 + 0.368i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ 0.929 + 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.670192118 + 0.5099851752i\)
\(L(\frac12)\) \(\approx\) \(2.670192118 + 0.5099851752i\)
\(L(1)\) \(\approx\) \(1.383534936 + 0.1982280496i\)
\(L(1)\) \(\approx\) \(1.383534936 + 0.1982280496i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.599 + 0.800i)T \)
5 \( 1 + (0.540 - 0.841i)T \)
7 \( 1 + (-0.977 + 0.212i)T \)
11 \( 1 + (0.841 - 0.540i)T \)
13 \( 1 + (-0.800 + 0.599i)T \)
17 \( 1 + (0.755 - 0.654i)T \)
19 \( 1 + (0.479 + 0.877i)T \)
23 \( 1 + (0.877 - 0.479i)T \)
29 \( 1 + (-0.212 - 0.977i)T \)
31 \( 1 + (0.877 + 0.479i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (-0.800 - 0.599i)T \)
43 \( 1 + (-0.212 + 0.977i)T \)
47 \( 1 + (0.989 - 0.142i)T \)
53 \( 1 + (0.989 + 0.142i)T \)
59 \( 1 + (-0.599 + 0.800i)T \)
61 \( 1 + (0.349 + 0.936i)T \)
67 \( 1 + (0.142 - 0.989i)T \)
71 \( 1 + (0.540 + 0.841i)T \)
73 \( 1 + (0.959 - 0.281i)T \)
79 \( 1 + (0.281 + 0.959i)T \)
83 \( 1 + (0.997 + 0.0713i)T \)
97 \( 1 + (0.841 + 0.540i)T \)
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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

−22.37780438404288188702369051769, −21.72167168017126347172847118395, −20.47640721208497792851216492042, −19.76748199862190828339464103102, −19.12680013203822435986083394799, −18.469385689275899482754048963814, −17.33109134087258705093369039086, −17.089503173210988551475625229519, −15.41481105509063869292567124364, −14.90470572605059354005007136358, −14.01510011840567185137092356113, −13.31655593181493294400890904250, −12.525512018711986201129604125, −11.71884373731622678231227967665, −10.40379286775190331860603502316, −9.69044831981211057307387680842, −8.97603207380515175654766564272, −7.638440286043888616671459928147, −6.9416681364546404416894819805, −6.411958448712514583455479171524, −5.24890161222760032691329140509, −3.56036522868442854385492245368, −3.02860224049835847723491117874, −1.98947665401014939838095034154, −0.811631924266992619930382980725, 0.79638824618765616524942740961, 2.19518583246114908200759353981, 3.19975380096101890880545193806, 4.12489415693199874195015154320, 5.132977057855433734489965058361, 5.946536376148702703923175967191, 7.11549189712562376202585292781, 8.353386218326872532020068563427, 9.16848884493997213106456279739, 9.63226574855276957045197648551, 10.369519633734282284005868642350, 11.80046556847676872735416196681, 12.42248503268387525159288590999, 13.65345961647916549537166125912, 14.03985181415375828724386540159, 15.062990719222645644394460855762, 16.07424823091715020237700219400, 16.65851765203276254596161422569, 17.09928074881947017437625236554, 18.65517263470746411297856491081, 19.36248840914231116737979069322, 20.011607694281665997270806608184, 21.01456869970828642251824946724, 21.398341137048771610605003597277, 22.39913728897877169100122219953

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