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Properties

Label 1-93-93.53-r0-0-0
Degree $1$
Conductor $93$
Sign $0.836 + 0.547i$
Analytic cond. $0.431890$
Root an. cond. $0.431890$
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  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.5 − 0.866i)5-s + (0.669 − 0.743i)7-s + (−0.309 + 0.951i)8-s + (0.913 − 0.406i)10-s + (−0.978 − 0.207i)11-s + (0.104 + 0.994i)13-s + (0.978 − 0.207i)14-s + (−0.809 + 0.587i)16-s + (−0.978 + 0.207i)17-s + (−0.104 + 0.994i)19-s + (0.978 + 0.207i)20-s + (−0.669 − 0.743i)22-s + (0.309 − 0.951i)23-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.5 − 0.866i)5-s + (0.669 − 0.743i)7-s + (−0.309 + 0.951i)8-s + (0.913 − 0.406i)10-s + (−0.978 − 0.207i)11-s + (0.104 + 0.994i)13-s + (0.978 − 0.207i)14-s + (−0.809 + 0.587i)16-s + (−0.978 + 0.207i)17-s + (−0.104 + 0.994i)19-s + (0.978 + 0.207i)20-s + (−0.669 − 0.743i)22-s + (0.309 − 0.951i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.836 + 0.547i$
Analytic conductor: \(0.431890\)
Root analytic conductor: \(0.431890\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 93,\ (0:\ ),\ 0.836 + 0.547i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.523125452 + 0.4537641106i\)
\(L(\frac12)\) \(\approx\) \(1.523125452 + 0.4537641106i\)
\(L(1)\) \(\approx\) \(1.537199591 + 0.3660029774i\)
\(L(1)\) \(\approx\) \(1.537199591 + 0.3660029774i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.669 - 0.743i)T \)
11 \( 1 + (-0.978 - 0.207i)T \)
13 \( 1 + (0.104 + 0.994i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.913 + 0.406i)T \)
43 \( 1 + (0.104 - 0.994i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (0.669 + 0.743i)T \)
59 \( 1 + (-0.913 - 0.406i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (0.978 + 0.207i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (0.913 - 0.406i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−30.35739341825618181302124044602, −29.32857716654059779888445620714, −28.38592063015081848470200616541, −27.31342209740568947546402459183, −25.873260999266685892984039898995, −24.82143643960604562590368036610, −23.73975745891265357074690518139, −22.588976814334557766767913685114, −21.76909130634122779069584786989, −20.93382701239015653867951009387, −19.73710479128407304181341253943, −18.38380753065803880352181583540, −17.79878641616615214286355850895, −15.49437753284112363002225236418, −15.04785479300917405692706726693, −13.69674674347550861198501976511, −12.81171236064751566158990875741, −11.297836614311235576213401545175, −10.64333892665087826292125724911, −9.25059997181007299189853805144, −7.37718873702291980372669483540, −5.88296287925795163305820251897, −4.94330829961589826841892390967, −3.05511062450002322870617818401, −2.09173072549554982202064660839, 2.0541083182628835925708111083, 4.11182048606224726407463350766, 5.02292336726405060942509364145, 6.33549307807126493397667623078, 7.77810271743830968347964119507, 8.8068830583480872623242739724, 10.61354436396221906423759626797, 11.96920124703120040367342736038, 13.23879260148506881421151746868, 13.86039250270524669049023408009, 15.15829561495480701695959657154, 16.50861417274069824705195418524, 17.05303056918190961438232343136, 18.37712912225754350316383229505, 20.3597346817528003077766635586, 20.90395169734541684646764643031, 21.89906755398611564929691813012, 23.366866795565133083597079199025, 24.04810587638689457698115936280, 24.85727664642138749727093822302, 26.131919001634166379998546032100, 26.93711885876574928684168428567, 28.60376121842243971252600145828, 29.40729261328285717566418552094, 30.6873019957601967796558308058

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