Properties

Label 1-547-547.292-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.996 + 0.0815i$
Analytic cond. $2.54025$
Root an. cond. $2.54025$
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  + (0.770 − 0.636i)2-s + (0.623 − 0.781i)3-s + (0.188 − 0.982i)4-s + (−0.479 + 0.877i)5-s + (−0.0172 − 0.999i)6-s + (−0.418 − 0.908i)7-s + (−0.479 − 0.877i)8-s + (−0.222 − 0.974i)9-s + (0.188 + 0.982i)10-s + (−0.354 + 0.935i)11-s + (−0.650 − 0.759i)12-s + (−0.900 − 0.433i)13-s + (−0.900 − 0.433i)14-s + (0.386 + 0.922i)15-s + (−0.928 − 0.370i)16-s + (−0.832 + 0.553i)17-s + ⋯
L(s)  = 1  + (0.770 − 0.636i)2-s + (0.623 − 0.781i)3-s + (0.188 − 0.982i)4-s + (−0.479 + 0.877i)5-s + (−0.0172 − 0.999i)6-s + (−0.418 − 0.908i)7-s + (−0.479 − 0.877i)8-s + (−0.222 − 0.974i)9-s + (0.188 + 0.982i)10-s + (−0.354 + 0.935i)11-s + (−0.650 − 0.759i)12-s + (−0.900 − 0.433i)13-s + (−0.900 − 0.433i)14-s + (0.386 + 0.922i)15-s + (−0.928 − 0.370i)16-s + (−0.832 + 0.553i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.996 + 0.0815i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (292, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.996 + 0.0815i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06259606236 - 1.532563700i\)
\(L(\frac12)\) \(\approx\) \(-0.06259606236 - 1.532563700i\)
\(L(1)\) \(\approx\) \(0.9589282603 - 1.004275078i\)
\(L(1)\) \(\approx\) \(0.9589282603 - 1.004275078i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.770 - 0.636i)T \)
3 \( 1 + (0.623 - 0.781i)T \)
5 \( 1 + (-0.479 + 0.877i)T \)
7 \( 1 + (-0.418 - 0.908i)T \)
11 \( 1 + (-0.354 + 0.935i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
17 \( 1 + (-0.832 + 0.553i)T \)
19 \( 1 + (-0.154 - 0.987i)T \)
23 \( 1 + (0.322 - 0.946i)T \)
29 \( 1 + (0.940 + 0.338i)T \)
31 \( 1 + (-0.289 - 0.957i)T \)
37 \( 1 + (-0.994 + 0.103i)T \)
41 \( 1 + T \)
43 \( 1 + (0.915 + 0.402i)T \)
47 \( 1 + (0.120 - 0.992i)T \)
53 \( 1 + (0.386 + 0.922i)T \)
59 \( 1 + (0.568 - 0.822i)T \)
61 \( 1 + (0.449 + 0.893i)T \)
67 \( 1 + (-0.650 - 0.759i)T \)
71 \( 1 + (0.675 - 0.736i)T \)
73 \( 1 + (0.386 - 0.922i)T \)
79 \( 1 + (0.915 + 0.402i)T \)
83 \( 1 + (-0.748 - 0.663i)T \)
89 \( 1 + (0.449 - 0.893i)T \)
97 \( 1 + (-0.0862 + 0.996i)T \)
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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

−23.98096942459962981828246669234, −22.878284820799941804341022777649, −22.093116279180381390948919087205, −21.29434790588847422404859911423, −20.88519812350193232263393784186, −19.67382793742395590459274058059, −19.126612430912605911679513796832, −17.610293737218996929021938922288, −16.58339865111586681541654266632, −15.898303649644136186084505240587, −15.609398337960637229842245222, −14.50683622798687697317628884952, −13.75640255321693682820619488453, −12.802304890903185358862595393572, −12.018342870263013692101446245974, −11.10796897163089923267594822940, −9.59437315177796526637106381267, −8.78773056316179456827621421481, −8.23051591356821226729154646807, −7.14796606786581541835375351803, −5.71140425049277016802478680583, −5.097392885291139198299579469881, −4.15811363039912746976653851915, −3.22301954034249576563167745695, −2.280664861287368480746629027073, 0.52673294487983370951198632821, 2.21008364707740895854173439242, 2.765880254549667094076927852094, 3.851474554741137675722633069170, 4.707733600961270332834099420830, 6.37299289454624718799326753715, 6.96984818681635612221228823504, 7.68284957533483479326001423906, 9.16037805613401249997728333241, 10.25005762847845063529228443871, 10.83379385049003328084939744813, 12.0249898001601800466957913867, 12.76497354084805417677827956142, 13.4288219222015128279098584050, 14.378005165103078629925198112043, 14.99612113126067152461132252982, 15.69110262247744438381706954944, 17.36837380493215349357558075121, 18.12881753549897100534510749518, 19.12964690569402131928337874431, 19.79783063767971764203862184331, 20.11046847777036837388864448215, 21.19893234538670030153280610782, 22.46024330818715293566438046738, 22.780049796930568290384981822361

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