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Properties

Label 1-3381-3381.1025-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.952 - 0.305i$
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  + (0.973 − 0.229i)2-s + (0.894 − 0.446i)4-s + (−0.966 − 0.255i)5-s + (0.768 − 0.639i)8-s + (−0.999 − 0.0271i)10-s + (0.00679 − 0.999i)11-s + (−0.101 + 0.994i)13-s + (0.601 − 0.798i)16-s + (−0.833 + 0.552i)17-s + (−0.723 + 0.690i)19-s + (−0.979 + 0.202i)20-s + (−0.222 − 0.974i)22-s + (0.869 + 0.494i)25-s + (0.128 + 0.991i)26-s + (0.0611 + 0.998i)29-s + ⋯
L(s)  = 1  + (0.973 − 0.229i)2-s + (0.894 − 0.446i)4-s + (−0.966 − 0.255i)5-s + (0.768 − 0.639i)8-s + (−0.999 − 0.0271i)10-s + (0.00679 − 0.999i)11-s + (−0.101 + 0.994i)13-s + (0.601 − 0.798i)16-s + (−0.833 + 0.552i)17-s + (−0.723 + 0.690i)19-s + (−0.979 + 0.202i)20-s + (−0.222 − 0.974i)22-s + (0.869 + 0.494i)25-s + (0.128 + 0.991i)26-s + (0.0611 + 0.998i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.952 - 0.305i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.952 - 0.305i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.605481103 - 0.4080988627i\)
\(L(\frac12)\) \(\approx\) \(2.605481103 - 0.4080988627i\)
\(L(1)\) \(\approx\) \(1.617897046 - 0.2798582848i\)
\(L(1)\) \(\approx\) \(1.617897046 - 0.2798582848i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.973 - 0.229i)T \)
5 \( 1 + (-0.966 - 0.255i)T \)
11 \( 1 + (0.00679 - 0.999i)T \)
13 \( 1 + (-0.101 + 0.994i)T \)
17 \( 1 + (-0.833 + 0.552i)T \)
19 \( 1 + (-0.723 + 0.690i)T \)
29 \( 1 + (0.0611 + 0.998i)T \)
31 \( 1 + (0.995 - 0.0950i)T \)
37 \( 1 + (0.994 - 0.108i)T \)
41 \( 1 + (0.262 + 0.965i)T \)
43 \( 1 + (-0.768 - 0.639i)T \)
47 \( 1 + (0.365 + 0.930i)T \)
53 \( 1 + (0.984 - 0.175i)T \)
59 \( 1 + (0.999 + 0.0271i)T \)
61 \( 1 + (0.923 - 0.384i)T \)
67 \( 1 + (0.928 - 0.371i)T \)
71 \( 1 + (-0.996 + 0.0815i)T \)
73 \( 1 + (0.612 - 0.790i)T \)
79 \( 1 + (0.0475 + 0.998i)T \)
83 \( 1 + (-0.999 - 0.0407i)T \)
89 \( 1 + (-0.546 - 0.837i)T \)
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

−19.07584266331917482085303694750, −17.97064258523653787473046848828, −17.45765082031723356520034953618, −16.61436892703035426880594584842, −15.71630074110238049817301033960, −15.3077527798600371719399596228, −14.94958210528588594622793986407, −14.051175462529503918708754532573, −13.11558687513277080296756351094, −12.7912169585321740557441284923, −11.79707990813493292885065563254, −11.50128447215800411032653757778, −10.58297462328348006347827431490, −9.94152667444076864747024392016, −8.6469476590329666402990664457, −8.05138529855218402546649525915, −7.19221488001095298765552990653, −6.84356063735044097958839915761, −5.89067779003479862563771978880, −4.90432504046663716632062115152, −4.40238655798945286772660509446, −3.72725252221037505310124766350, −2.66617279688153472690744329836, −2.29075515277742292633684106467, −0.70464903754866458755253795547, 0.83910478380930006013491814272, 1.82989958759422999916735935789, 2.77263150352558643752933116125, 3.64874673460561321434903143459, 4.20995259312362313436295630926, 4.81462802375303973294777381683, 5.83255173091311288905672212886, 6.5164884937785575846741049856, 7.16207820326362144492690015983, 8.20581436635558620212400367944, 8.66682044372486507653213186251, 9.7719179441027363370516044659, 10.72291714172846809366900983623, 11.27688260027662146960367588709, 11.77068302047804630997109709423, 12.58347747836624750749972476405, 13.124857545470493096338189854276, 13.94743845698070519096883759065, 14.611326191783710316514603865444, 15.21108144895922309477843387573, 16.00649946677725518811701428078, 16.47345483517654674832715445680, 17.075392459268239485526327481040, 18.3996362632139089508661850278, 19.0627358750645301288605284080

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