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Properties

Label 1-712-712.115-r0-0-0
Degree $1$
Conductor $712$
Sign $0.953 + 0.302i$
Analytic cond. $3.30651$
Root an. cond. $3.30651$
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  + (−0.936 + 0.349i)3-s + (−0.989 − 0.142i)5-s + (−0.800 + 0.599i)7-s + (0.755 − 0.654i)9-s + (0.142 + 0.989i)11-s + (−0.349 − 0.936i)13-s + (0.977 − 0.212i)15-s + (−0.540 − 0.841i)17-s + (−0.997 − 0.0713i)19-s + (0.540 − 0.841i)21-s + (0.0713 − 0.997i)23-s + (0.959 + 0.281i)25-s + (−0.479 + 0.877i)27-s + (−0.599 − 0.800i)29-s + (0.0713 + 0.997i)31-s + ⋯
L(s)  = 1  + (−0.936 + 0.349i)3-s + (−0.989 − 0.142i)5-s + (−0.800 + 0.599i)7-s + (0.755 − 0.654i)9-s + (0.142 + 0.989i)11-s + (−0.349 − 0.936i)13-s + (0.977 − 0.212i)15-s + (−0.540 − 0.841i)17-s + (−0.997 − 0.0713i)19-s + (0.540 − 0.841i)21-s + (0.0713 − 0.997i)23-s + (0.959 + 0.281i)25-s + (−0.479 + 0.877i)27-s + (−0.599 − 0.800i)29-s + (0.0713 + 0.997i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.953 + 0.302i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ 0.953 + 0.302i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5035222209 + 0.07788832377i\)
\(L(\frac12)\) \(\approx\) \(0.5035222209 + 0.07788832377i\)
\(L(1)\) \(\approx\) \(0.5543492862 + 0.06141859553i\)
\(L(1)\) \(\approx\) \(0.5543492862 + 0.06141859553i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (-0.936 + 0.349i)T \)
5 \( 1 + (-0.989 - 0.142i)T \)
7 \( 1 + (-0.800 + 0.599i)T \)
11 \( 1 + (0.142 + 0.989i)T \)
13 \( 1 + (-0.349 - 0.936i)T \)
17 \( 1 + (-0.540 - 0.841i)T \)
19 \( 1 + (-0.997 - 0.0713i)T \)
23 \( 1 + (0.0713 - 0.997i)T \)
29 \( 1 + (-0.599 - 0.800i)T \)
31 \( 1 + (0.0713 + 0.997i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-0.349 + 0.936i)T \)
43 \( 1 + (0.599 - 0.800i)T \)
47 \( 1 + (-0.909 + 0.415i)T \)
53 \( 1 + (0.909 + 0.415i)T \)
59 \( 1 + (0.936 + 0.349i)T \)
61 \( 1 + (0.877 + 0.479i)T \)
67 \( 1 + (0.415 - 0.909i)T \)
71 \( 1 + (0.989 - 0.142i)T \)
73 \( 1 + (0.654 - 0.755i)T \)
79 \( 1 + (0.755 + 0.654i)T \)
83 \( 1 + (0.977 + 0.212i)T \)
97 \( 1 + (-0.142 + 0.989i)T \)
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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

−22.583789555860862641226394196, −21.98958660331550349043979159985, −21.1272147567300331061945906550, −19.71521007718419029929060069145, −19.25223925410752032239835157890, −18.76885489998998975424269807542, −17.4927103176021835157702567085, −16.74130719249644976481929957040, −16.23159662308454956982728002337, −15.38848760878804840636755360706, −14.27556374921421258465392900218, −13.19688201625498999700195994654, −12.63384425859900609971644611020, −11.55905856535999580525679166014, −11.08058125549391006814886882971, −10.24593923390021970579128655472, −9.03396342879426959475875179715, −7.983571436256987564488095992890, −6.9790818247830629516462565346, −6.51055690840340966549378632325, −5.42273827370943554925079956492, −4.130943120552055583462857316734, −3.64549557037954193975484457204, −2.009860666889017533711042753904, −0.563530670020375040651193630746, 0.55534230298386248850417630892, 2.38146790772751517777343431803, 3.568769480706251558180152736686, 4.56729161546574080248974967513, 5.231650197272031667307320196012, 6.49869924035990744384292971957, 7.05145732546949107608539797833, 8.27086797376186046131929567445, 9.29313949297835467636383667461, 10.15400604076058914027977723029, 10.964167268353421554764862670852, 12.024097813759443384732498770018, 12.400638009170586716685584076296, 13.17536541640888362432667775735, 14.947390028979382579247364508148, 15.28064206330348099775248715541, 16.06338085293763764083939724092, 16.81322815497607113046593778803, 17.72677239391477713463233175506, 18.50916514932065814848447290085, 19.394356921143087109662981546011, 20.21187064988965756002657068709, 21.005847680451595123567661708336, 22.19809571788940730452294976638, 22.705799527272476016728268041904

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