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Properties

Label 1-1323-1323.58-r0-0-0
Degree $1$
Conductor $1323$
Sign $0.817 + 0.576i$
Analytic cond. $6.14398$
Root an. cond. $6.14398$
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  + (−0.583 + 0.811i)2-s + (−0.318 − 0.947i)4-s + (0.921 − 0.388i)5-s + (0.955 + 0.294i)8-s + (−0.222 + 0.974i)10-s + (−0.583 + 0.811i)11-s + (−0.969 + 0.246i)13-s + (−0.797 + 0.603i)16-s + (0.623 − 0.781i)17-s + 19-s + (−0.661 − 0.749i)20-s + (−0.318 − 0.947i)22-s + (0.980 + 0.198i)23-s + (0.698 − 0.715i)25-s + (0.365 − 0.930i)26-s + ⋯
L(s)  = 1  + (−0.583 + 0.811i)2-s + (−0.318 − 0.947i)4-s + (0.921 − 0.388i)5-s + (0.955 + 0.294i)8-s + (−0.222 + 0.974i)10-s + (−0.583 + 0.811i)11-s + (−0.969 + 0.246i)13-s + (−0.797 + 0.603i)16-s + (0.623 − 0.781i)17-s + 19-s + (−0.661 − 0.749i)20-s + (−0.318 − 0.947i)22-s + (0.980 + 0.198i)23-s + (0.698 − 0.715i)25-s + (0.365 − 0.930i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.817 + 0.576i$
Analytic conductor: \(6.14398\)
Root analytic conductor: \(6.14398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (0:\ ),\ 0.817 + 0.576i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.226042762 + 0.3885712274i\)
\(L(\frac12)\) \(\approx\) \(1.226042762 + 0.3885712274i\)
\(L(1)\) \(\approx\) \(0.8996960141 + 0.2428725273i\)
\(L(1)\) \(\approx\) \(0.8996960141 + 0.2428725273i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.583 + 0.811i)T \)
5 \( 1 + (0.921 - 0.388i)T \)
11 \( 1 + (-0.583 + 0.811i)T \)
13 \( 1 + (-0.969 + 0.246i)T \)
17 \( 1 + (0.623 - 0.781i)T \)
19 \( 1 + T \)
23 \( 1 + (0.980 + 0.198i)T \)
29 \( 1 + (0.980 - 0.198i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (0.365 + 0.930i)T \)
41 \( 1 + (-0.797 - 0.603i)T \)
43 \( 1 + (-0.124 - 0.992i)T \)
47 \( 1 + (0.995 + 0.0995i)T \)
53 \( 1 + (-0.988 + 0.149i)T \)
59 \( 1 + (0.456 + 0.889i)T \)
61 \( 1 + (-0.318 + 0.947i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.365 - 0.930i)T \)
73 \( 1 + (0.826 + 0.563i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (-0.969 - 0.246i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−20.96595011779146300224723055388, −20.02951732461581027645436020280, −19.33239293428494149387237749814, −18.508334877300371005423219117573, −18.05381340158546150371834894847, −17.13419152384512144872279963116, −16.681138438443725890202631452014, −15.66099093214334985581827801045, −14.45687945526173092070044398966, −13.9217990344973163913386793478, −12.90114342833091689246993121839, −12.5027575952737404399227786833, −11.288556861190411573921111339496, −10.72876067995453535006988486946, −9.94415979803354657904039670487, −9.391389070815998199567685094616, −8.4058788370349710618264482794, −7.61423332445404452260256203864, −6.72375478193574631257789512119, −5.56278988567680371874425648843, −4.84934240122785907862698166087, −3.36628618554690349420850632418, −2.897384775760884821073395111704, −1.90983844893134535644341341295, −0.87462789111435400085595300289, 0.84967282695807452899888666695, 1.89799624164411604625784600209, 2.85202406858444159211452971719, 4.59141413299341332435743864452, 5.15625483573667838173798147108, 5.79043212476386729493560258719, 7.059966941010927401085348885614, 7.35000167584670979419869336995, 8.49372912668775025728408630975, 9.38234743148784982394597082978, 9.84003811490116381000651928300, 10.494178778296260476744454995677, 11.75046495903853963738241459210, 12.649708963951279375068703484906, 13.60252395040460866689322534113, 14.13639175788630661360230451306, 15.01674008598906453163526546004, 15.72442475899991602921008683567, 16.685350326796152059412457256821, 17.0962976807400580032267547634, 17.94253715674734538039889515543, 18.444629214936592439882234495133, 19.32899181822112372904011888596, 20.342935068875659961693700251051, 20.74648013100852841478779400778

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