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Properties

Label 1-93-93.35-r1-0-0
Degree $1$
Conductor $93$
Sign $-0.962 - 0.272i$
Analytic cond. $9.99423$
Root an. cond. $9.99423$
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 − 5-s + (0.309 − 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)10-s + (−0.309 + 0.951i)11-s + (−0.809 − 0.587i)13-s + (−0.309 − 0.951i)14-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)17-s + (−0.809 + 0.587i)19-s + (−0.309 + 0.951i)20-s + (0.309 + 0.951i)22-s + (−0.309 − 0.951i)23-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s − 5-s + (0.309 − 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)10-s + (−0.309 + 0.951i)11-s + (−0.809 − 0.587i)13-s + (−0.309 − 0.951i)14-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)17-s + (−0.809 + 0.587i)19-s + (−0.309 + 0.951i)20-s + (0.309 + 0.951i)22-s + (−0.309 − 0.951i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $-0.962 - 0.272i$
Analytic conductor: \(9.99423\)
Root analytic conductor: \(9.99423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 93,\ (1:\ ),\ -0.962 - 0.272i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2117278763 - 1.522140266i\)
\(L(\frac12)\) \(\approx\) \(0.2117278763 - 1.522140266i\)
\(L(1)\) \(\approx\) \(0.9744156035 - 0.7657456851i\)
\(L(1)\) \(\approx\) \(0.9744156035 - 0.7657456851i\)

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 \( 1 - T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + (-0.809 + 0.587i)T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
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

−30.87301096205470370638254630380, −29.7775907088786412202362170305, −28.43682869296893973346113854373, −27.17389355185098223740412522161, −26.28907889262979571451114405231, −25.03446677708701456731013427259, −23.96860669401641412292703627402, −23.5222097696444383066182639855, −21.84830447849924177521914282236, −21.59044676748706542960457256543, −19.90360337556932633196227365945, −18.855271586897183992976906173093, −17.38024204269128137603352130849, −16.163587759226997977639497409513, −15.30932445084680541135186023861, −14.45767020176574430748185036307, −12.99317323635622426844214257249, −11.957597848857206207244771979735, −11.098155965375070170133966955726, −8.79283601229418723345312326785, −7.948251359740133851845165345117, −6.56426947808325974665951756622, −5.23508275934411807465472997963, −4.008573198177315621668012701242, −2.57278179217215558889835221240, 0.51761886023558058688815486414, 2.513841128693606930019636769835, 4.10419479255439940516388238445, 4.852771002406388990369246248785, 6.79547243277419683160423871449, 7.88618453908766399812142401901, 9.90763634185072394150234590596, 10.859344470363630044941548152655, 12.03878729020545402276677115327, 12.92217963902169558821366630764, 14.33386507726911774424591477749, 15.15960768882328661478623136630, 16.35436256098893573360439489023, 17.9134292435166218728413492416, 19.300718216318262907113189065835, 20.17745646773656065048668224433, 20.82242402903664383951404564641, 22.46646605992306710734389740716, 23.09567083783639499363281420929, 23.959744481515626145127559412593, 25.07408871839895363252989687237, 26.78802992084667469747402271674, 27.53604404523842797792547661722, 28.677892049186044734148995546200, 29.84805232990670249959660221835

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