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Properties

Label 1-3381-3381.593-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.426 + 0.904i$
Analytic cond. $15.7012$
Root an. cond. $15.7012$
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  + (−0.963 − 0.268i)2-s + (0.855 + 0.517i)4-s + (0.906 + 0.421i)5-s + (−0.685 − 0.728i)8-s + (−0.760 − 0.649i)10-s + (0.984 + 0.175i)11-s + (−0.882 + 0.470i)13-s + (0.464 + 0.885i)16-s + (−0.876 − 0.482i)17-s + (−0.580 + 0.814i)19-s + (0.557 + 0.830i)20-s + (−0.900 − 0.433i)22-s + (0.644 + 0.764i)25-s + (0.976 − 0.215i)26-s + (−0.0203 − 0.999i)29-s + ⋯
L(s)  = 1  + (−0.963 − 0.268i)2-s + (0.855 + 0.517i)4-s + (0.906 + 0.421i)5-s + (−0.685 − 0.728i)8-s + (−0.760 − 0.649i)10-s + (0.984 + 0.175i)11-s + (−0.882 + 0.470i)13-s + (0.464 + 0.885i)16-s + (−0.876 − 0.482i)17-s + (−0.580 + 0.814i)19-s + (0.557 + 0.830i)20-s + (−0.900 − 0.433i)22-s + (0.644 + 0.764i)25-s + (0.976 − 0.215i)26-s + (−0.0203 − 0.999i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.426 + 0.904i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.426 + 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9572988363 + 0.6072003535i\)
\(L(\frac12)\) \(\approx\) \(0.9572988363 + 0.6072003535i\)
\(L(1)\) \(\approx\) \(0.8104926164 + 0.08358764392i\)
\(L(1)\) \(\approx\) \(0.8104926164 + 0.08358764392i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.963 - 0.268i)T \)
5 \( 1 + (0.906 + 0.421i)T \)
11 \( 1 + (0.984 + 0.175i)T \)
13 \( 1 + (-0.882 + 0.470i)T \)
17 \( 1 + (-0.876 - 0.482i)T \)
19 \( 1 + (-0.580 + 0.814i)T \)
29 \( 1 + (-0.0203 - 0.999i)T \)
31 \( 1 + (0.786 + 0.618i)T \)
37 \( 1 + (-0.951 - 0.307i)T \)
41 \( 1 + (-0.818 + 0.574i)T \)
43 \( 1 + (0.685 - 0.728i)T \)
47 \( 1 + (0.955 - 0.294i)T \)
53 \( 1 + (0.115 - 0.993i)T \)
59 \( 1 + (0.760 + 0.649i)T \)
61 \( 1 + (0.675 - 0.737i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (0.523 + 0.852i)T \)
73 \( 1 + (-0.128 - 0.991i)T \)
79 \( 1 + (-0.327 + 0.945i)T \)
83 \( 1 + (0.488 + 0.872i)T \)
89 \( 1 + (0.777 + 0.628i)T \)
97 \( 1 + (0.959 + 0.281i)T \)
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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

−18.690840771142174093549783119554, −17.578507260603354373951171186470, −17.46368977639827918147903205610, −16.93249028179447124673231618818, −16.095748634453193846625033316591, −15.30733081283428519316144165586, −14.68600635990863204712645746248, −13.96861558520779540826351782025, −13.12784909374757511068282828370, −12.32630563277754592101031944290, −11.6276167665487943658088618016, −10.65496162082874184021641315612, −10.25621377854525141015766370705, −9.263904258818113060577608055626, −8.97800627871876473746988632349, −8.26051688857660838613472745331, −7.21627648047083294401185682720, −6.601468302322952453769266593778, −5.97576437094071274848551022511, −5.13468208696615223561310553779, −4.33619767697553743565055007021, −2.98638498662123740974823191253, −2.206401865391993353559099833245, −1.49188007630002628173944503209, −0.50410499823799128973664796201, 0.9764927479837237564386189708, 2.08745188243459320373010032082, 2.27535446621747054991576093225, 3.445681253755129695168431745662, 4.29893068620364584520918300771, 5.40236977463111693492720441270, 6.44428124176203840781287174175, 6.74657965517979454198075052252, 7.51622628544621343202182541989, 8.60731618927477524551531653401, 9.082479528168915754105158145200, 9.91111635455319163461196104366, 10.2277067065653515456416208177, 11.16586279100625718126633573658, 11.85840951720902407660310021755, 12.42494384492658694151811759817, 13.40555212503050948288379570432, 14.14794705568595671581951894090, 14.83081319012482165059930507221, 15.5587827741084669509852049583, 16.485044858173455233419324217218, 17.17316883197378752334396715267, 17.457972983767796232690713807016, 18.18985675582494524772819068640, 19.0624117182065786941478077859

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