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Properties

Label 1-3381-3381.2042-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: $\chi_{3381} (2042, \cdot )$
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

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

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