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Properties

Label 1-935-935.54-r0-0-0
Degree $1$
Conductor $935$
Sign $0.563 + 0.825i$
Analytic cond. $4.34212$
Root an. cond. $4.34212$
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  + (0.707 − 0.707i)2-s + (−0.923 − 0.382i)3-s − i·4-s + (−0.923 + 0.382i)6-s + (−0.382 − 0.923i)7-s + (−0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.382 + 0.923i)12-s + i·13-s + (−0.923 − 0.382i)14-s − 16-s + 18-s + (−0.707 + 0.707i)19-s + i·21-s + (−0.923 + 0.382i)23-s + (0.382 + 0.923i)24-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.923 − 0.382i)3-s − i·4-s + (−0.923 + 0.382i)6-s + (−0.382 − 0.923i)7-s + (−0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.382 + 0.923i)12-s + i·13-s + (−0.923 − 0.382i)14-s − 16-s + 18-s + (−0.707 + 0.707i)19-s + i·21-s + (−0.923 + 0.382i)23-s + (0.382 + 0.923i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(935\)    =    \(5 \cdot 11 \cdot 17\)
Sign: $0.563 + 0.825i$
Analytic conductor: \(4.34212\)
Root analytic conductor: \(4.34212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{935} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 935,\ (0:\ ),\ 0.563 + 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2415606099 + 0.1275507461i\)
\(L(\frac12)\) \(\approx\) \(0.2415606099 + 0.1275507461i\)
\(L(1)\) \(\approx\) \(0.7008979357 - 0.4271806469i\)
\(L(1)\) \(\approx\) \(0.7008979357 - 0.4271806469i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.923 - 0.382i)T \)
7 \( 1 + (-0.382 - 0.923i)T \)
13 \( 1 + iT \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (-0.923 + 0.382i)T \)
29 \( 1 + (-0.382 + 0.923i)T \)
31 \( 1 + (-0.923 - 0.382i)T \)
37 \( 1 + (-0.923 - 0.382i)T \)
41 \( 1 + (0.382 + 0.923i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.382 - 0.923i)T \)
67 \( 1 + T \)
71 \( 1 + (0.923 + 0.382i)T \)
73 \( 1 + (-0.382 + 0.923i)T \)
79 \( 1 + (0.923 - 0.382i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.382 + 0.923i)T \)
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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

−22.14173413044460171403921974588, −21.20070912074031624114195473864, −20.53234085037644205226667275870, −19.25941181622279043794187584397, −18.184470862542139670057070427607, −17.62723501849523723998686106495, −16.85851971538135620775502319901, −15.94632501314739666663375988015, −15.48111250877672593034060380221, −14.86444513570689638636552523675, −13.71069872917875457800518071187, −12.60490269998611230168548505450, −12.4180344285459735270022712206, −11.38448725590574695357673724567, −10.52985641019885075484923731857, −9.41016946127754109590520607362, −8.57568409990701837654949716773, −7.50861839984674916981515667589, −6.50519174531100296432560132522, −5.83992034698318636157922969692, −5.23154328333355273736125711502, −4.268963624648970679447629069340, −3.32756247893986745659498398019, −2.216490318738810246040595277529, −0.10411797849395118245159536756, 1.32584295696600982609858267007, 2.07886041125987404778478743547, 3.62410689378137446026525396448, 4.24476691144312612246577698338, 5.20812748160759772539204323210, 6.18738227185389820531186336280, 6.77201852722662062939843742685, 7.74855735283752225898946814434, 9.2902553553083633685258299084, 10.10165524959672146064843763869, 10.87772581974764685879438002598, 11.442386005988446096772062634974, 12.46106006580641226596459679032, 12.88767061834085743733950626667, 13.89070755137871978773452105119, 14.39827376628582301452645884960, 15.719390151316154172084127467147, 16.447673446350724476632698505034, 17.15818805407722672639289615199, 18.244360711558043635594889816555, 18.87040681665683396141881406728, 19.65805792910660364818449306663, 20.38409525612321114563798897346, 21.436935245259801395352210173672, 21.91027469881920576690615565743

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