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Properties

Label 1-3381-3381.101-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.199 + 0.979i$
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  + (0.984 + 0.175i)2-s + (0.938 + 0.346i)4-s + (−0.612 − 0.790i)5-s + (0.862 + 0.505i)8-s + (−0.464 − 0.885i)10-s + (−0.963 − 0.268i)11-s + (0.591 + 0.806i)13-s + (0.760 + 0.649i)16-s + (−0.169 + 0.985i)17-s + (−0.580 − 0.814i)19-s + (−0.301 − 0.953i)20-s + (−0.900 − 0.433i)22-s + (−0.248 + 0.968i)25-s + (0.440 + 0.897i)26-s + (0.768 + 0.639i)29-s + ⋯
L(s)  = 1  + (0.984 + 0.175i)2-s + (0.938 + 0.346i)4-s + (−0.612 − 0.790i)5-s + (0.862 + 0.505i)8-s + (−0.464 − 0.885i)10-s + (−0.963 − 0.268i)11-s + (0.591 + 0.806i)13-s + (0.760 + 0.649i)16-s + (−0.169 + 0.985i)17-s + (−0.580 − 0.814i)19-s + (−0.301 − 0.953i)20-s + (−0.900 − 0.433i)22-s + (−0.248 + 0.968i)25-s + (0.440 + 0.897i)26-s + (0.768 + 0.639i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.199 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.199 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.199 + 0.979i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.199 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.831203843 + 1.495959489i\)
\(L(\frac12)\) \(\approx\) \(1.831203843 + 1.495959489i\)
\(L(1)\) \(\approx\) \(1.599903469 + 0.2789004838i\)
\(L(1)\) \(\approx\) \(1.599903469 + 0.2789004838i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.984 + 0.175i)T \)
5 \( 1 + (-0.612 - 0.790i)T \)
11 \( 1 + (-0.963 - 0.268i)T \)
13 \( 1 + (0.591 + 0.806i)T \)
17 \( 1 + (-0.169 + 0.985i)T \)
19 \( 1 + (-0.580 - 0.814i)T \)
29 \( 1 + (0.768 + 0.639i)T \)
31 \( 1 + (0.786 - 0.618i)T \)
37 \( 1 + (-0.352 + 0.935i)T \)
41 \( 1 + (-0.377 + 0.925i)T \)
43 \( 1 + (-0.862 + 0.505i)T \)
47 \( 1 + (-0.733 - 0.680i)T \)
53 \( 1 + (-0.704 + 0.709i)T \)
59 \( 1 + (0.464 + 0.885i)T \)
61 \( 1 + (0.997 - 0.0679i)T \)
67 \( 1 + (-0.888 - 0.458i)T \)
71 \( 1 + (0.992 - 0.122i)T \)
73 \( 1 + (-0.314 - 0.949i)T \)
79 \( 1 + (-0.327 - 0.945i)T \)
83 \( 1 + (-0.0611 + 0.998i)T \)
89 \( 1 + (-0.427 + 0.903i)T \)
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

−18.822586573072471458413397893054, −18.10670612754301029625205600820, −17.34681754912250487353533513664, −16.063629139901955295515029901396, −15.81764285676280701106451140779, −15.25304595335168392060628071545, −14.39876346593562687556987956144, −13.92315378646706964960248679552, −13.047912689676285501842809443070, −12.48916057004563495940773165006, −11.67922324470864157520799356473, −11.116598070437526350136373742377, −10.27517170786447138599613223604, −10.054282927488587784159039316699, −8.46579989742745303720879825191, −7.874986393122777995270047986421, −7.09239193764768102796316526342, −6.46746470128008558981774061955, −5.60929727903424956114450746409, −4.90044168131514799865471677837, −4.05291326067734902883114409124, −3.27714828570336020855673489532, −2.70821191386045526356168242073, −1.87647882198976845668012853477, −0.48217449581742736002637081419, 1.14954549110042817910635863287, 2.050699686826926142353839644515, 3.05469663966464052017704952807, 3.775846664519260050042760735608, 4.70422751866520389464097273121, 4.915252409854280071630485444229, 6.09671791081975640382050638912, 6.58297162599926965757694806756, 7.567068065223647811952359849472, 8.35816757582090184003354230011, 8.67880863324810065023345951067, 9.99549319977908367313504614043, 10.8159996921469008250399236277, 11.47323853793385086176009751877, 12.04108503385327502181218185153, 12.97934710180269792906706046747, 13.225567216271387930381205618983, 13.965908214269546358690995960511, 15.064869774386994082079448448346, 15.356468518474122713028659707105, 16.154595140546479027946480607716, 16.634998347742474592833771394475, 17.31865792286089141999705642927, 18.31353988103457192125352036762, 19.26617464709890948204049855422

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