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Properties

Label 1-517-517.173-r1-0-0
Degree $1$
Conductor $517$
Sign $-0.239 - 0.970i$
Analytic cond. $55.5593$
Root an. cond. $55.5593$
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  + (−0.994 − 0.109i)2-s + (−0.981 − 0.190i)3-s + (0.976 + 0.216i)4-s + (0.554 + 0.832i)5-s + (0.955 + 0.295i)6-s + (−0.256 − 0.966i)7-s + (−0.946 − 0.321i)8-s + (0.927 + 0.373i)9-s + (−0.460 − 0.887i)10-s + (−0.917 − 0.398i)12-s + (0.435 − 0.900i)13-s + (0.149 + 0.988i)14-s + (−0.385 − 0.922i)15-s + (0.905 + 0.423i)16-s + (0.999 + 0.0273i)17-s + (−0.881 − 0.472i)18-s + ⋯
L(s)  = 1  + (−0.994 − 0.109i)2-s + (−0.981 − 0.190i)3-s + (0.976 + 0.216i)4-s + (0.554 + 0.832i)5-s + (0.955 + 0.295i)6-s + (−0.256 − 0.966i)7-s + (−0.946 − 0.321i)8-s + (0.927 + 0.373i)9-s + (−0.460 − 0.887i)10-s + (−0.917 − 0.398i)12-s + (0.435 − 0.900i)13-s + (0.149 + 0.988i)14-s + (−0.385 − 0.922i)15-s + (0.905 + 0.423i)16-s + (0.999 + 0.0273i)17-s + (−0.881 − 0.472i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $-0.239 - 0.970i$
Analytic conductor: \(55.5593\)
Root analytic conductor: \(55.5593\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (1:\ ),\ -0.239 - 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5165570473 - 0.6597766419i\)
\(L(\frac12)\) \(\approx\) \(0.5165570473 - 0.6597766419i\)
\(L(1)\) \(\approx\) \(0.5905576963 - 0.1395014626i\)
\(L(1)\) \(\approx\) \(0.5905576963 - 0.1395014626i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.994 - 0.109i)T \)
3 \( 1 + (-0.981 - 0.190i)T \)
5 \( 1 + (0.554 + 0.832i)T \)
7 \( 1 + (-0.256 - 0.966i)T \)
13 \( 1 + (0.435 - 0.900i)T \)
17 \( 1 + (0.999 + 0.0273i)T \)
19 \( 1 + (0.937 - 0.347i)T \)
23 \( 1 + (0.203 - 0.979i)T \)
29 \( 1 + (-0.881 - 0.472i)T \)
31 \( 1 + (-0.484 - 0.874i)T \)
37 \( 1 + (0.149 - 0.988i)T \)
41 \( 1 + (0.0136 + 0.999i)T \)
43 \( 1 + (0.917 - 0.398i)T \)
53 \( 1 + (0.598 + 0.800i)T \)
59 \( 1 + (0.507 + 0.861i)T \)
61 \( 1 + (-0.986 - 0.163i)T \)
67 \( 1 + (-0.775 + 0.631i)T \)
71 \( 1 + (0.994 - 0.109i)T \)
73 \( 1 + (0.531 - 0.847i)T \)
79 \( 1 + (-0.758 + 0.652i)T \)
83 \( 1 + (-0.792 + 0.609i)T \)
89 \( 1 + (0.962 - 0.269i)T \)
97 \( 1 + (0.905 - 0.423i)T \)
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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

−23.931818552842036259981384098000, −22.77209491410795461242655011782, −21.55807027092443950348359477705, −21.22435122184251865334470456973, −20.25461068708215340011681435430, −19.00627358832972353301283747436, −18.42029612600300655579994378200, −17.634968721382887074580361592574, −16.76708188893825389098753403373, −16.200643859345552917464369890664, −15.58411114666425039326308920307, −14.31980359359193716724344614607, −12.91739428964202245337510859706, −12.03021687317477797533447245899, −11.53312417170916687797559546672, −10.32865827449508805184851585353, −9.4382622561026098592612446592, −9.02525170742076981283541821801, −7.72826542533018751685846434971, −6.602436268887188256729022199772, −5.67535011757879630965220347755, −5.207679545256089042772453847, −3.47654773138676810946043906222, −1.83690374905559950305571031459, −1.07918499146754282052637591432, 0.41373450800744652059024165705, 1.28867060794147889607308949952, 2.68342727222292907577564764159, 3.81819404166280650183261264109, 5.57714577198438707267526230454, 6.2329545803079303518516117448, 7.30191181714728625364361651434, 7.69110096120137271527193099785, 9.39970182542497902889489534940, 10.18763750071245002494736160882, 10.74331016517814983847930190875, 11.436852084286981337121162738512, 12.604532942182682659438427314636, 13.45997950928521028658782656496, 14.69213081897437873095573008067, 15.74741824835352700565066265942, 16.681084639355959210408323439764, 17.17363956802944600630616443674, 18.17171061515438763039649655469, 18.475962137395667027478737168510, 19.53329081121560502319864807148, 20.54573413160039805917991397042, 21.30975031201635131643421411218, 22.46733746966798513664664922493, 22.91001506596052222106196195694

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