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Properties

Label 1-92-92.71-r1-0-0
Degree $1$
Conductor $92$
Sign $0.818 + 0.574i$
Analytic cond. $9.88677$
Root an. cond. $9.88677$
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  + (0.959 − 0.281i)3-s + (0.841 + 0.540i)5-s + (0.142 + 0.989i)7-s + (0.841 − 0.540i)9-s + (−0.415 + 0.909i)11-s + (−0.142 + 0.989i)13-s + (0.959 + 0.281i)15-s + (−0.654 − 0.755i)17-s + (0.654 − 0.755i)19-s + (0.415 + 0.909i)21-s + (0.415 + 0.909i)25-s + (0.654 − 0.755i)27-s + (−0.654 − 0.755i)29-s + (0.959 + 0.281i)31-s + (−0.142 + 0.989i)33-s + ⋯
L(s)  = 1  + (0.959 − 0.281i)3-s + (0.841 + 0.540i)5-s + (0.142 + 0.989i)7-s + (0.841 − 0.540i)9-s + (−0.415 + 0.909i)11-s + (−0.142 + 0.989i)13-s + (0.959 + 0.281i)15-s + (−0.654 − 0.755i)17-s + (0.654 − 0.755i)19-s + (0.415 + 0.909i)21-s + (0.415 + 0.909i)25-s + (0.654 − 0.755i)27-s + (−0.654 − 0.755i)29-s + (0.959 + 0.281i)31-s + (−0.142 + 0.989i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $0.818 + 0.574i$
Analytic conductor: \(9.88677\)
Root analytic conductor: \(9.88677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{92} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 92,\ (1:\ ),\ 0.818 + 0.574i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.523006399 + 0.7966350128i\)
\(L(\frac12)\) \(\approx\) \(2.523006399 + 0.7966350128i\)
\(L(1)\) \(\approx\) \(1.664099662 + 0.2521117494i\)
\(L(1)\) \(\approx\) \(1.664099662 + 0.2521117494i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.959 - 0.281i)T \)
5 \( 1 + (0.841 + 0.540i)T \)
7 \( 1 + (0.142 + 0.989i)T \)
11 \( 1 + (-0.415 + 0.909i)T \)
13 \( 1 + (-0.142 + 0.989i)T \)
17 \( 1 + (-0.654 - 0.755i)T \)
19 \( 1 + (0.654 - 0.755i)T \)
29 \( 1 + (-0.654 - 0.755i)T \)
31 \( 1 + (0.959 + 0.281i)T \)
37 \( 1 + (0.841 - 0.540i)T \)
41 \( 1 + (0.841 + 0.540i)T \)
43 \( 1 + (0.959 - 0.281i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.142 - 0.989i)T \)
59 \( 1 + (0.142 - 0.989i)T \)
61 \( 1 + (-0.959 - 0.281i)T \)
67 \( 1 + (-0.415 - 0.909i)T \)
71 \( 1 + (-0.415 - 0.909i)T \)
73 \( 1 + (-0.654 + 0.755i)T \)
79 \( 1 + (0.142 - 0.989i)T \)
83 \( 1 + (-0.841 + 0.540i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (0.841 + 0.540i)T \)
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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

−29.93239438232885163740810799687, −29.19440051361174748545954306948, −27.72396988350447265098785155938, −26.68938852942633904760851517241, −25.913009682354216833916646826, −24.74493282143295341821378972551, −24.04726744485761339423450611624, −22.389925627801686658151261049857, −21.22534851963113908415751161053, −20.48692882765145820113135742720, −19.60780051837873686584273702507, −18.17987389020366398679441067369, −16.96756712718466724494575439007, −15.93906995347050715877957503903, −14.541958912639368466023490877822, −13.57277494497270203483748466697, −12.88541105029446172888816194415, −10.745365566031093916485268046980, −9.90376655919993617854976701064, −8.60994566067675723214570423952, −7.62830084641279140414470106759, −5.817269803417478752286115829095, −4.34908874496239890165935393865, −2.924150089646087674392178613138, −1.23746508226654160280972314067, 1.98042502384307438314399426252, 2.75136560435352112906962809071, 4.74051861251371966470145629458, 6.41370946766264094628624189655, 7.53344283775477472208613367789, 9.12692205686240558372928941263, 9.70107500645393315357999743643, 11.475758308505580219410687963669, 12.85426869861500656411058124790, 13.88557314628971299800971584066, 14.84087730331706567540233617347, 15.80080426788559980902456430469, 17.72556587424257395687483195701, 18.36582283556494309418560589046, 19.42898132289104528389860054543, 20.75891081527963857783071198869, 21.53245277151877106951315835109, 22.65174715332329774873541664593, 24.28476240068823914147401737343, 25.01806784720667927479242910591, 25.983288588464404052300477418634, 26.67012432308277566942206885036, 28.31393177594578368841766404099, 29.16986738220082955205471296950, 30.46753561247343296462977602824

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