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Properties

Label 1-712-712.235-r1-0-0
Degree $1$
Conductor $712$
Sign $0.976 - 0.213i$
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  + (−0.841 + 0.540i)3-s + (0.654 + 0.755i)5-s + (−0.654 − 0.755i)7-s + (0.415 − 0.909i)9-s + (−0.654 + 0.755i)11-s + (0.841 − 0.540i)13-s + (−0.959 − 0.281i)15-s + (−0.959 + 0.281i)17-s + (−0.415 + 0.909i)19-s + (0.959 + 0.281i)21-s + (0.415 − 0.909i)23-s + (−0.142 + 0.989i)25-s + (0.142 + 0.989i)27-s + (−0.654 − 0.755i)29-s + (0.415 + 0.909i)31-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)3-s + (0.654 + 0.755i)5-s + (−0.654 − 0.755i)7-s + (0.415 − 0.909i)9-s + (−0.654 + 0.755i)11-s + (0.841 − 0.540i)13-s + (−0.959 − 0.281i)15-s + (−0.959 + 0.281i)17-s + (−0.415 + 0.909i)19-s + (0.959 + 0.281i)21-s + (0.415 − 0.909i)23-s + (−0.142 + 0.989i)25-s + (0.142 + 0.989i)27-s + (−0.654 − 0.755i)29-s + (0.415 + 0.909i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.976 - 0.213i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ 0.976 - 0.213i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.088044530 - 0.1175493293i\)
\(L(\frac12)\) \(\approx\) \(1.088044530 - 0.1175493293i\)
\(L(1)\) \(\approx\) \(0.7835706709 + 0.1356637403i\)
\(L(1)\) \(\approx\) \(0.7835706709 + 0.1356637403i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (-0.841 + 0.540i)T \)
5 \( 1 + (0.654 + 0.755i)T \)
7 \( 1 + (-0.654 - 0.755i)T \)
11 \( 1 + (-0.654 + 0.755i)T \)
13 \( 1 + (0.841 - 0.540i)T \)
17 \( 1 + (-0.959 + 0.281i)T \)
19 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (0.415 - 0.909i)T \)
29 \( 1 + (-0.654 - 0.755i)T \)
31 \( 1 + (0.415 + 0.909i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.841 - 0.540i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (-0.841 - 0.540i)T \)
53 \( 1 + (-0.841 + 0.540i)T \)
59 \( 1 + (-0.841 - 0.540i)T \)
61 \( 1 + (-0.142 - 0.989i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (0.654 - 0.755i)T \)
73 \( 1 + (0.415 + 0.909i)T \)
79 \( 1 + (-0.415 - 0.909i)T \)
83 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + (-0.654 - 0.755i)T \)
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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

−22.30699707827561244682218382983, −21.6800951133708854612475994084, −21.07395462166430798164417079618, −19.86361631698128915290148482845, −19.00641034292541022120661117570, −18.274694780554000803972564534032, −17.63085468286041715455334817921, −16.61660100158025123116443178267, −16.11900007846535802747573624148, −15.31001325457002133417802671765, −13.68318502759782822877075335046, −13.20303432944047518712908606981, −12.70590211194675390255028347570, −11.45369118440648434095019887146, −11.046637736589156943268506094914, −9.69474809657354170505025951989, −8.975271731349720686825715189967, −8.08615353110209523375001832665, −6.73372273411594768085855344054, −6.08425090628271567522534353781, −5.375619371886528670534928884011, −4.462535634809414300730890478724, −2.86060005368908446786762101535, −1.85288613936587304031508420396, −0.71874453563516374852232061615, 0.40437355520868037115202499100, 1.882145129634026981503702518707, 3.201399392229556825848927284836, 4.093057376538006587026556316559, 5.13594986561219392491173428789, 6.27899497457182207827074840000, 6.59357769030696751903245014779, 7.77497079442236914397349407733, 9.16003860466354319623584161388, 10.13541217323269586732907578137, 10.516054524557360941382953082034, 11.17746913605671676290371380476, 12.57139030315906348035130887302, 13.11107269521930067216026435716, 14.13129108596736464350234861503, 15.20837504981278468080613149280, 15.73050444984660155387183940070, 16.82026228699856884875764144022, 17.37811568580094842484318810404, 18.21943525777832171577554850662, 18.83736029044286655589954199527, 20.22462392613291478334231505084, 20.787171304917226011372725801344, 21.66203578874977132007146109725, 22.56907493549827436498828170890

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