Properties

Label 1-1520-1520.373-r0-0-0
Degree $1$
Conductor $1520$
Sign $0.00303 + 0.999i$
Analytic cond. $7.05885$
Root an. cond. $7.05885$
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  + (0.5 − 0.866i)3-s i·7-s + (−0.5 − 0.866i)9-s + i·11-s + (0.5 + 0.866i)13-s + (0.866 + 0.5i)17-s + (0.866 + 0.5i)21-s + (−0.866 + 0.5i)23-s − 27-s + (−0.866 + 0.5i)29-s − 31-s + (0.866 + 0.5i)33-s − 37-s + 39-s + (−0.5 + 0.866i)41-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s i·7-s + (−0.5 − 0.866i)9-s + i·11-s + (0.5 + 0.866i)13-s + (0.866 + 0.5i)17-s + (0.866 + 0.5i)21-s + (−0.866 + 0.5i)23-s − 27-s + (−0.866 + 0.5i)29-s − 31-s + (0.866 + 0.5i)33-s − 37-s + 39-s + (−0.5 + 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00303 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00303 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.00303 + 0.999i$
Analytic conductor: \(7.05885\)
Root analytic conductor: \(7.05885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1520,\ (0:\ ),\ 0.00303 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8475940433 + 0.8501676815i\)
\(L(\frac12)\) \(\approx\) \(0.8475940433 + 0.8501676815i\)
\(L(1)\) \(\approx\) \(1.081157288 + 0.06911132131i\)
\(L(1)\) \(\approx\) \(1.081157288 + 0.06911132131i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 - iT \)
11 \( 1 + iT \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (-0.866 + 0.5i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−20.53998165787117056307679616686, −19.838955661788706691192837052214, −19.05028571509850410805326670062, −18.22775739645881680213782881449, −17.19527904389621273297515533853, −16.48220445184589130134529244333, −16.055326496167200367582629898493, −15.13922762147117704399533506453, −14.28647791293087315347391447646, −13.7741840492381739084999245381, −13.08905542851879831408998399619, −11.919966059740869910831320977386, −10.87132273896842443723094680050, −10.56167064247983142996701780674, −9.70047392703028984499413531732, −8.866761350110051034317570076715, −7.99148871564180940753430479892, −7.48194748018943796226514620721, −6.12088335334397642013498620384, −5.40841122323762991082426002451, −4.42481237253942547751876868158, −3.50221047781984284043869906675, −3.144909865461443716782919121110, −1.73152904292367200284995550051, −0.38520763549932232696608758330, 1.70996661845773882986743331595, 1.82503753180007970312172704601, 3.119973457511297930730281996104, 3.90633644199136373101888014599, 5.19168377657960884861668552467, 6.00377787754044149804255113090, 6.79201914379377033064862345312, 7.628002501996757226698612105355, 8.363700266780097264124045159416, 9.20319014183693690063726344845, 9.73999735200388132044466266797, 11.03204556066975716272686613117, 11.9511374507938519388359024742, 12.386019410105299399741292427260, 13.11182550482759673257510925440, 14.06596481996984374520958547699, 14.68664932314649729230461152273, 15.33525949612361969176995149102, 16.25685728202274476584979887390, 17.204735864151892635833469760242, 18.05154647422392206809628278772, 18.5370666648102331882847470322, 19.186749738247782046006659468136, 19.94958033386577979491819944457, 20.730286118487185364325305316761

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