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Properties

Label 1-3381-3381.752-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.740 + 0.671i$
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  + (−0.288 − 0.957i)2-s + (−0.833 + 0.552i)4-s + (0.704 + 0.709i)5-s + (0.768 + 0.639i)8-s + (0.476 − 0.879i)10-s + (−0.869 − 0.494i)11-s + (−0.101 − 0.994i)13-s + (0.390 − 0.920i)16-s + (0.894 − 0.446i)17-s + (−0.235 + 0.971i)19-s + (−0.979 − 0.202i)20-s + (−0.222 + 0.974i)22-s + (−0.00679 + 0.999i)25-s + (−0.923 + 0.384i)26-s + (0.0611 − 0.998i)29-s + ⋯
L(s)  = 1  + (−0.288 − 0.957i)2-s + (−0.833 + 0.552i)4-s + (0.704 + 0.709i)5-s + (0.768 + 0.639i)8-s + (0.476 − 0.879i)10-s + (−0.869 − 0.494i)11-s + (−0.101 − 0.994i)13-s + (0.390 − 0.920i)16-s + (0.894 − 0.446i)17-s + (−0.235 + 0.971i)19-s + (−0.979 − 0.202i)20-s + (−0.222 + 0.974i)22-s + (−0.00679 + 0.999i)25-s + (−0.923 + 0.384i)26-s + (0.0611 − 0.998i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.740 + 0.671i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (752, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.740 + 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03153064746 - 0.08172191500i\)
\(L(\frac12)\) \(\approx\) \(-0.03153064746 - 0.08172191500i\)
\(L(1)\) \(\approx\) \(0.7138102643 - 0.2648383125i\)
\(L(1)\) \(\approx\) \(0.7138102643 - 0.2648383125i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.288 - 0.957i)T \)
5 \( 1 + (0.704 + 0.709i)T \)
11 \( 1 + (-0.869 - 0.494i)T \)
13 \( 1 + (-0.101 - 0.994i)T \)
17 \( 1 + (0.894 - 0.446i)T \)
19 \( 1 + (-0.235 + 0.971i)T \)
29 \( 1 + (0.0611 - 0.998i)T \)
31 \( 1 + (-0.580 + 0.814i)T \)
37 \( 1 + (-0.403 - 0.915i)T \)
41 \( 1 + (0.262 - 0.965i)T \)
43 \( 1 + (-0.768 + 0.639i)T \)
47 \( 1 + (-0.988 + 0.149i)T \)
53 \( 1 + (-0.644 + 0.764i)T \)
59 \( 1 + (-0.476 + 0.879i)T \)
61 \( 1 + (-0.128 - 0.991i)T \)
67 \( 1 + (-0.786 + 0.618i)T \)
71 \( 1 + (-0.996 - 0.0815i)T \)
73 \( 1 + (-0.990 + 0.135i)T \)
79 \( 1 + (-0.888 + 0.458i)T \)
83 \( 1 + (-0.999 + 0.0407i)T \)
89 \( 1 + (0.998 + 0.0543i)T \)
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

−18.974929984716332257876244242942, −18.44202556530296159476778003755, −17.667001203432507921698587392887, −17.17734839224343359750684275353, −16.39587289651697087669754124376, −16.09635719701105226300941764995, −15.011929536562189057366850612958, −14.61217358883180922193482505907, −13.6719333724888766667620867393, −13.17221810428678632859856253165, −12.580719365325741293767476255929, −11.58367710338242351472181052225, −10.492173957119389516436551628794, −9.92470416190058620024422571201, −9.267935689687527598458834753921, −8.63025855117651926940072744839, −7.9115729801136918022304096881, −7.10214674493266345767804616669, −6.39647690695680349359312740126, −5.622139842450933677509842877184, −4.87699891090224019939684137759, −4.48496036234459932641482725580, −3.22525754830604055715611599840, −1.96567608437178545957474080478, −1.32997649548807877333805534577, 0.02827529183667702686514807169, 1.289818244058590507780342929734, 2.11694027689954670179163999444, 3.07921769745842646140229474339, 3.26632438902788799181927419951, 4.498890820576330463860057635539, 5.55893514769550115394896206419, 5.82013975226411103461075110577, 7.22345167167732165709724771787, 7.81549773910505194559332772827, 8.53214196361069148807341224328, 9.479829632317352145243132306653, 10.17624631077576886833549953714, 10.50732033461861010021671764333, 11.21234536250523315853366704742, 12.10461556106409474636869911860, 12.80416541058185825754245290834, 13.38749715278839429087096866780, 14.16395598140089970969216337984, 14.63031355159220156753827710686, 15.699792262656636610259249172660, 16.49545850176326778905225093340, 17.28848115547283458546651153837, 17.90818964312592560806538258474, 18.46421346645704720982614201969

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