Properties

Label 1-712-712.549-r1-0-0
Degree $1$
Conductor $712$
Sign $0.0853 + 0.996i$
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  + (−0.349 + 0.936i)3-s + (0.989 − 0.142i)5-s + (−0.599 + 0.800i)7-s + (−0.755 − 0.654i)9-s + (−0.142 + 0.989i)11-s + (0.936 + 0.349i)13-s + (−0.212 + 0.977i)15-s + (0.540 − 0.841i)17-s + (−0.0713 − 0.997i)19-s + (−0.540 − 0.841i)21-s + (0.997 − 0.0713i)23-s + (0.959 − 0.281i)25-s + (0.877 − 0.479i)27-s + (0.800 + 0.599i)29-s + (0.997 + 0.0713i)31-s + ⋯
L(s)  = 1  + (−0.349 + 0.936i)3-s + (0.989 − 0.142i)5-s + (−0.599 + 0.800i)7-s + (−0.755 − 0.654i)9-s + (−0.142 + 0.989i)11-s + (0.936 + 0.349i)13-s + (−0.212 + 0.977i)15-s + (0.540 − 0.841i)17-s + (−0.0713 − 0.997i)19-s + (−0.540 − 0.841i)21-s + (0.997 − 0.0713i)23-s + (0.959 − 0.281i)25-s + (0.877 − 0.479i)27-s + (0.800 + 0.599i)29-s + (0.997 + 0.0713i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0853 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0853 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.0853 + 0.996i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ 0.0853 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.697771100 + 1.558595384i\)
\(L(\frac12)\) \(\approx\) \(1.697771100 + 1.558595384i\)
\(L(1)\) \(\approx\) \(1.106075448 + 0.4690104252i\)
\(L(1)\) \(\approx\) \(1.106075448 + 0.4690104252i\)

Euler product

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

−22.48010647515020946622930972624, −21.22690012457646200180506710145, −20.80441222877468405169958581112, −19.34713338117482919211758658266, −19.10217899698173652182469870186, −18.07293790796118361739113122229, −17.33615456743683512869103097091, −16.704175312718758824141042647557, −15.90540134600507634674552400565, −14.34387184021894746101283400171, −13.8732174374609377946633468201, −13.04619781513107985516322520067, −12.586206450796283326625819572880, −11.15789586105625040251745388413, −10.633938694697607669728170507586, −9.71081453778786085335768072023, −8.46387023371351766446743434119, −7.77120908442754126415117747619, −6.370803753838317214191525656486, −6.26698113918732643719213989861, −5.23598271586062794737339531778, −3.64524923758522035740363712126, −2.75450570301773044250067299594, −1.40869731587151507949390030349, −0.73338338657165958756876343406, 0.93799121036070266921324365622, 2.43614759059036270533277504849, 3.21210726723649500512544548245, 4.65602939025034406992198776916, 5.22725992119884424377509571286, 6.2050147346306213232835443223, 6.92428484968828627960342994198, 8.68458311104436793013189019339, 9.21074638303437833412349893777, 9.91690319994966875926635354341, 10.71194044488490127884515523062, 11.758284212480901479759777700205, 12.577477677869159260950488662988, 13.52818229740471229188383712330, 14.42714853139638366484967255271, 15.42717251615517698216222628912, 15.932650905447351384820316491778, 16.844937822201368698477078967056, 17.67465274463075721007923377837, 18.28682437457015493800732941815, 19.36273270543394354555244733091, 20.51758547117214904282455921439, 21.067861174625963325794587313443, 21.659689455448511796732994315953, 22.64247011270029057029889486897

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