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Properties

Label 1-3381-3381.614-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.755 + 0.655i$
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  + (0.00679 − 0.999i)2-s + (−0.999 − 0.0135i)4-s + (−0.802 − 0.596i)5-s + (−0.0203 + 0.999i)8-s + (−0.601 + 0.798i)10-s + (0.973 − 0.229i)11-s + (−0.947 + 0.320i)13-s + (0.999 + 0.0271i)16-s + (0.511 + 0.859i)17-s + (−0.723 − 0.690i)19-s + (0.794 + 0.607i)20-s + (−0.222 − 0.974i)22-s + (0.288 + 0.957i)25-s + (0.314 + 0.949i)26-s + (−0.488 + 0.872i)29-s + ⋯
L(s)  = 1  + (0.00679 − 0.999i)2-s + (−0.999 − 0.0135i)4-s + (−0.802 − 0.596i)5-s + (−0.0203 + 0.999i)8-s + (−0.601 + 0.798i)10-s + (0.973 − 0.229i)11-s + (−0.947 + 0.320i)13-s + (0.999 + 0.0271i)16-s + (0.511 + 0.859i)17-s + (−0.723 − 0.690i)19-s + (0.794 + 0.607i)20-s + (−0.222 − 0.974i)22-s + (0.288 + 0.957i)25-s + (0.314 + 0.949i)26-s + (−0.488 + 0.872i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.755 + 0.655i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (614, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.755 + 0.655i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1417661890 - 0.3794577020i\)
\(L(\frac12)\) \(\approx\) \(-0.1417661890 - 0.3794577020i\)
\(L(1)\) \(\approx\) \(0.6022697207 - 0.4118626357i\)
\(L(1)\) \(\approx\) \(0.6022697207 - 0.4118626357i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.00679 - 0.999i)T \)
5 \( 1 + (-0.802 - 0.596i)T \)
11 \( 1 + (0.973 - 0.229i)T \)
13 \( 1 + (-0.947 + 0.320i)T \)
17 \( 1 + (0.511 + 0.859i)T \)
19 \( 1 + (-0.723 - 0.690i)T \)
29 \( 1 + (-0.488 + 0.872i)T \)
31 \( 1 + (0.995 + 0.0950i)T \)
37 \( 1 + (-0.848 - 0.529i)T \)
41 \( 1 + (0.917 - 0.396i)T \)
43 \( 1 + (0.0203 + 0.999i)T \)
47 \( 1 + (-0.988 - 0.149i)T \)
53 \( 1 + (-0.963 + 0.268i)T \)
59 \( 1 + (0.601 - 0.798i)T \)
61 \( 1 + (-0.665 - 0.746i)T \)
67 \( 1 + (0.928 + 0.371i)T \)
71 \( 1 + (0.933 - 0.359i)T \)
73 \( 1 + (-0.906 + 0.421i)T \)
79 \( 1 + (0.0475 - 0.998i)T \)
83 \( 1 + (0.182 - 0.983i)T \)
89 \( 1 + (-0.694 - 0.719i)T \)
97 \( 1 + (0.654 - 0.755i)T \)
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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

−19.201175440684058768703282149553, −18.53496091955403308775748069740, −17.71924492354276637289627938304, −17.05722976428615943744537202282, −16.534922217069032256599737957396, −15.64826673574969802514716834108, −15.14249534663437145298597216358, −14.46804191691747913917344518432, −14.09629763834741662165535332021, −13.090655323152752195348359692112, −12.138259638945282998402418743234, −11.87025830169054465259498542793, −10.723798724261074778180416069803, −9.880607843079031056470438687190, −9.39667012342912057032614162878, −8.28963310305703193309321634600, −7.87065272222069347609647680793, −7.04121364902637382585071713494, −6.59544929062368375907825585826, −5.706304309076765426581172544115, −4.78474968280833731499527045761, −4.12515227456538551867647764826, −3.418533473416661792452924815427, −2.43903085917583537065238664341, −1.028812232206449390834772551946, 0.147568448989294791262112828261, 1.22750055245872367633967194915, 1.95487714957782874998408372333, 3.07826298681533115677047762968, 3.75119440260503564096196932826, 4.493424320418459310594234079626, 5.00884268192554309802584307524, 6.077572811130951225785255900978, 7.06517741450832818178701620118, 7.97919023062721709312460334167, 8.63253530127857903019794414785, 9.24729089314902692672796505884, 9.92192032559012785214850413123, 10.91711964965216468213747359572, 11.378255574607606439045399548127, 12.21701083338493550730390873358, 12.5528472664539311812260940858, 13.25486009075678525231140679090, 14.40889313627093998129022973386, 14.58234632401891064822359233553, 15.614116316785734781328911904041, 16.51520393578779243883691913942, 17.22101648385677902838111295441, 17.541847136339605906510825846870, 18.81090402135543043087282523055

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