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Properties

Label 1-3381-3381.524-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.307 + 0.951i$
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  + (0.523 + 0.852i)2-s + (−0.452 + 0.891i)4-s + (−0.0611 + 0.998i)5-s + (−0.996 + 0.0815i)8-s + (−0.882 + 0.470i)10-s + (0.992 − 0.122i)11-s + (−0.262 − 0.965i)13-s + (−0.591 − 0.806i)16-s + (−0.452 − 0.891i)17-s + (−0.415 − 0.909i)19-s + (−0.862 − 0.505i)20-s + (0.623 + 0.781i)22-s + (−0.992 − 0.122i)25-s + (0.685 − 0.728i)26-s + (0.452 + 0.891i)29-s + ⋯
L(s)  = 1  + (0.523 + 0.852i)2-s + (−0.452 + 0.891i)4-s + (−0.0611 + 0.998i)5-s + (−0.996 + 0.0815i)8-s + (−0.882 + 0.470i)10-s + (0.992 − 0.122i)11-s + (−0.262 − 0.965i)13-s + (−0.591 − 0.806i)16-s + (−0.452 − 0.891i)17-s + (−0.415 − 0.909i)19-s + (−0.862 − 0.505i)20-s + (0.623 + 0.781i)22-s + (−0.992 − 0.122i)25-s + (0.685 − 0.728i)26-s + (0.452 + 0.891i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.307 + 0.951i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.307 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.568618323 + 1.142171849i\)
\(L(\frac12)\) \(\approx\) \(1.568618323 + 1.142171849i\)
\(L(1)\) \(\approx\) \(1.090069864 + 0.6756022323i\)
\(L(1)\) \(\approx\) \(1.090069864 + 0.6756022323i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.523 + 0.852i)T \)
5 \( 1 + (-0.0611 + 0.998i)T \)
11 \( 1 + (0.992 - 0.122i)T \)
13 \( 1 + (-0.262 - 0.965i)T \)
17 \( 1 + (-0.452 - 0.891i)T \)
19 \( 1 + (-0.415 - 0.909i)T \)
29 \( 1 + (0.452 + 0.891i)T \)
31 \( 1 + (0.142 - 0.989i)T \)
37 \( 1 + (-0.377 + 0.925i)T \)
41 \( 1 + (-0.0611 + 0.998i)T \)
43 \( 1 + (0.996 + 0.0815i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (0.999 + 0.0407i)T \)
59 \( 1 + (0.882 - 0.470i)T \)
61 \( 1 + (-0.685 - 0.728i)T \)
67 \( 1 + (0.841 + 0.540i)T \)
71 \( 1 + (-0.101 - 0.994i)T \)
73 \( 1 + (0.768 + 0.639i)T \)
79 \( 1 + (-0.654 - 0.755i)T \)
83 \( 1 + (0.742 - 0.670i)T \)
89 \( 1 + (0.557 + 0.830i)T \)
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

−19.210500467865996424834448460237, −17.984974993313991811816459630028, −17.32010368006319951536699103853, −16.69418406847589714105940065225, −15.86627891848735159546924237885, −15.082558002914524550356332037603, −14.2496572708738859187689959615, −13.87369449599282618596383413891, −12.89348085289016411699119525740, −12.35754307502583861742667668611, −11.89644200522456835430131777046, −11.142908605771395541913341847872, −10.264342457431466099730441180507, −9.58155997725152393829644278385, −8.8428743778733378776396353958, −8.43488655241495137965707079217, −7.13539414917659736169195718244, −6.23508310069777344738755710757, −5.64448632787922409199783658222, −4.62139950258011564148772648245, −4.14247074608618583940445943074, −3.58675477338914966565980531866, −2.1769463521313698148443807838, −1.72577203750158207774992739617, −0.81027635377691783316008684468, 0.63117276127524590107271361773, 2.27988995684804586796692319267, 3.00756624555752060406490123053, 3.65617046403684435343729865350, 4.58853552981824946570573001759, 5.265601136476044972351335359433, 6.32645887986749795471037290143, 6.64954558687421880315835143685, 7.40134639906072335947300599195, 8.10397455269890066503192076633, 8.97827867762311294991317759457, 9.660532268152269905269954305, 10.59460945467044680246248770330, 11.48429630308244167411989034702, 11.90711308523172477895788124566, 12.98692338171473263232491815371, 13.48132355783433951718462508043, 14.329039291155497520892457159056, 14.761341288369497438668652413835, 15.42844858831988150258541337274, 15.98697379773221331641838465152, 16.93412783347040145924935132981, 17.5149363766549160287043163367, 18.0816101578753488254239399346, 18.75610926216768722168342606026

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