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Properties

Label 1-5520-5520.4829-r0-0-0
Degree $1$
Conductor $5520$
Sign $-0.923 - 0.382i$
Analytic cond. $25.6347$
Root an. cond. $25.6347$
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  − 7-s − i·11-s − i·13-s − 17-s − i·19-s − i·29-s + 31-s − i·37-s + 41-s − i·43-s + 47-s + 49-s + i·53-s + i·59-s − i·61-s + ⋯
L(s)  = 1  − 7-s − i·11-s − i·13-s − 17-s − i·19-s − i·29-s + 31-s − i·37-s + 41-s − i·43-s + 47-s + 49-s + i·53-s + i·59-s − i·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.923 - 0.382i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (4829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ -0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1823365676 - 0.9166678275i\)
\(L(\frac12)\) \(\approx\) \(0.1823365676 - 0.9166678275i\)
\(L(1)\) \(\approx\) \(0.8302591914 - 0.2566598649i\)
\(L(1)\) \(\approx\) \(0.8302591914 - 0.2566598649i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 - T \)
29 \( 1 \)
31 \( 1 \)
37 \( 1 - iT \)
41 \( 1 \)
43 \( 1 - iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 - T \)
67 \( 1 \)
71 \( 1 - iT \)
73 \( 1 \)
79 \( 1 \)
83 \( 1 \)
89 \( 1 \)
97 \( 1 \)
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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

−18.25060125060815816751278615670, −17.43866854959395101427898834906, −16.7983980475340703557492849556, −16.17441342283410299344930272223, −15.591094837408842136675932106168, −14.91363588876445130945125908796, −14.14636405551623279964975214915, −13.555584137890684057535382164314, −12.70321652323602268794770469598, −12.36602134218174568582405278927, −11.550610132375295147666115729776, −10.7981652112010482704402249259, −9.94465430308331038315446839587, −9.570122976675874008109020522729, −8.842609765434535128149715365891, −8.05451697301686470753499391195, −7.12248241477511602833127593104, −6.63788094557039594301959595577, −6.07629665187881760889291673061, −5.02334220627393431445764689383, −4.331420178001757408343009676747, −3.70313156829834407774369027568, −2.746471179648924834392746283776, −2.05349528853175634330821454303, −1.16509944842897096787992910128, 0.29553581535151486239942900130, 0.919142961321938077940390073148, 2.47774948428377192876693299920, 2.73229558338175487830161241011, 3.719182223388624192096968835894, 4.33722602261833112283284756914, 5.41104814882735304217539929966, 5.96752701037680346810198266673, 6.62099248418840221783891865765, 7.36787354331312158228083204671, 8.17557949300762658192829706453, 8.93638238954179186622362686358, 9.39276049944476900047507079590, 10.35569869713990322461548223255, 10.804729498720496618342218216232, 11.55348463829341462567323843150, 12.35888133649834092420768259340, 13.06824974916392663920300042593, 13.51152915948301855655294852457, 14.10192626616870743835467290118, 15.19627197910363274412892354013, 15.70974535877044890881876678312, 16.00966503783867387935517856025, 17.14179126931188459602513297460, 17.35528946793976196406792152060

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