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Properties

Label 1-1027-1027.1017-r1-0-0
Degree $1$
Conductor $1027$
Sign $0.889 - 0.457i$
Analytic cond. $110.366$
Root an. cond. $110.366$
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  + (−0.200 − 0.979i)2-s + (−0.428 − 0.903i)3-s + (−0.919 + 0.391i)4-s + (−0.970 + 0.239i)5-s + (−0.799 + 0.600i)6-s + (0.996 + 0.0804i)7-s + (0.568 + 0.822i)8-s + (−0.632 + 0.774i)9-s + (0.428 + 0.903i)10-s + (0.692 − 0.721i)11-s + (0.748 + 0.663i)12-s + (−0.120 − 0.992i)14-s + (0.632 + 0.774i)15-s + (0.692 − 0.721i)16-s + (0.919 − 0.391i)17-s + (0.885 + 0.464i)18-s + ⋯
L(s)  = 1  + (−0.200 − 0.979i)2-s + (−0.428 − 0.903i)3-s + (−0.919 + 0.391i)4-s + (−0.970 + 0.239i)5-s + (−0.799 + 0.600i)6-s + (0.996 + 0.0804i)7-s + (0.568 + 0.822i)8-s + (−0.632 + 0.774i)9-s + (0.428 + 0.903i)10-s + (0.692 − 0.721i)11-s + (0.748 + 0.663i)12-s + (−0.120 − 0.992i)14-s + (0.632 + 0.774i)15-s + (0.692 − 0.721i)16-s + (0.919 − 0.391i)17-s + (0.885 + 0.464i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1027\)    =    \(13 \cdot 79\)
Sign: $0.889 - 0.457i$
Analytic conductor: \(110.366\)
Root analytic conductor: \(110.366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1027} (1017, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1027,\ (1:\ ),\ 0.889 - 0.457i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9425986827 - 0.2284771309i\)
\(L(\frac12)\) \(\approx\) \(0.9425986827 - 0.2284771309i\)
\(L(1)\) \(\approx\) \(0.6026622054 - 0.4150936066i\)
\(L(1)\) \(\approx\) \(0.6026622054 - 0.4150936066i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.200 - 0.979i)T \)
3 \( 1 + (-0.428 - 0.903i)T \)
5 \( 1 + (-0.970 + 0.239i)T \)
7 \( 1 + (0.996 + 0.0804i)T \)
11 \( 1 + (0.692 - 0.721i)T \)
17 \( 1 + (0.919 - 0.391i)T \)
19 \( 1 + (-0.0402 - 0.999i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.632 + 0.774i)T \)
31 \( 1 + (-0.748 + 0.663i)T \)
37 \( 1 + (0.845 - 0.534i)T \)
41 \( 1 + (-0.278 + 0.960i)T \)
43 \( 1 + (-0.692 - 0.721i)T \)
47 \( 1 + (-0.885 + 0.464i)T \)
53 \( 1 + (-0.568 - 0.822i)T \)
59 \( 1 + (-0.799 + 0.600i)T \)
61 \( 1 + (0.0402 + 0.999i)T \)
67 \( 1 + (0.948 - 0.316i)T \)
71 \( 1 + (-0.428 + 0.903i)T \)
73 \( 1 + (0.120 + 0.992i)T \)
83 \( 1 + (0.120 - 0.992i)T \)
89 \( 1 + (0.428 + 0.903i)T \)
97 \( 1 + (-0.0402 - 0.999i)T \)
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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

−21.60429439725483893379451248393, −20.611513486525908536987211014277, −20.03494839075253035941367930444, −18.90164068784068225465113027675, −18.15338419646075375061940471906, −17.176897919123783968246197331235, −16.75176170896213033621174675270, −16.05027653659291427573129738780, −15.01833139193295640708136442372, −14.820445416832205544768908635956, −14.03278913006978810946147517086, −12.4920115751846862619169703932, −11.93448078614200849454772792739, −10.94801485201768946716046443439, −10.07741064068413870229616544212, −9.31796060744206203952513078987, −8.19696558393118740100284012698, −7.92612248823158307112415157220, −6.69492134859255889363238850487, −5.78913532591483754512490572433, −4.80837257511548953521266204850, −4.291659190127572833626343203336, −3.563940058718595102785555685106, −1.50128177543602258829500035862, −0.331975658072608327057874788149, 0.83798113320996246084157141200, 1.48504294653952092929665853605, 2.734458866036754351554223877778, 3.58135246734675089515334100077, 4.71372602381622489698530783643, 5.47325835618793590601012336459, 6.83416510976216536805858789494, 7.71724072369259070608457134705, 8.29556791573475033420527294913, 9.11746516851053111214643581719, 10.48419214580203863054924018696, 11.348008736976947396834005394, 11.56620213576317209596957869892, 12.28323572016834430690695237416, 13.21575581415078347708301248714, 14.15916320878569205304840991625, 14.6254471162366832726159522545, 16.09393095805211874332142980453, 16.871189331147745961458172047356, 17.79212769952973027850266260315, 18.28894892099432462440559130989, 19.04363153176251707689417850569, 19.75533239776708788739684496827, 20.18491903436164573574888811968, 21.5344208193653601600811984125

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