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Properties

Label 1-5520-5520.3947-r0-0-0
Degree $1$
Conductor $5520$
Sign $0.755 + 0.655i$
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  + (−0.755 + 0.654i)7-s + (0.540 − 0.841i)11-s + (0.654 − 0.755i)13-s + (−0.909 + 0.415i)17-s + (0.909 + 0.415i)19-s + (0.909 − 0.415i)29-s + (0.142 + 0.989i)31-s + (−0.959 − 0.281i)37-s + (−0.959 + 0.281i)41-s + (−0.142 + 0.989i)43-s − i·47-s + (0.142 − 0.989i)49-s + (−0.654 − 0.755i)53-s + (0.755 + 0.654i)59-s + (0.989 − 0.142i)61-s + ⋯
L(s)  = 1  + (−0.755 + 0.654i)7-s + (0.540 − 0.841i)11-s + (0.654 − 0.755i)13-s + (−0.909 + 0.415i)17-s + (0.909 + 0.415i)19-s + (0.909 − 0.415i)29-s + (0.142 + 0.989i)31-s + (−0.959 − 0.281i)37-s + (−0.959 + 0.281i)41-s + (−0.142 + 0.989i)43-s − i·47-s + (0.142 − 0.989i)49-s + (−0.654 − 0.755i)53-s + (0.755 + 0.654i)59-s + (0.989 − 0.142i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.755 + 0.655i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (3947, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ 0.755 + 0.655i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.438295132 + 0.5369988427i\)
\(L(\frac12)\) \(\approx\) \(1.438295132 + 0.5369988427i\)
\(L(1)\) \(\approx\) \(1.023494582 + 0.07457250282i\)
\(L(1)\) \(\approx\) \(1.023494582 + 0.07457250282i\)

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 \( 1 + (-0.755 + 0.654i)T \)
11 \( 1 + (0.540 - 0.841i)T \)
13 \( 1 + (0.654 - 0.755i)T \)
17 \( 1 + (-0.909 + 0.415i)T \)
19 \( 1 + (0.909 + 0.415i)T \)
29 \( 1 + (0.909 - 0.415i)T \)
31 \( 1 + (0.142 + 0.989i)T \)
37 \( 1 + (-0.959 - 0.281i)T \)
41 \( 1 + (-0.959 + 0.281i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.654 - 0.755i)T \)
59 \( 1 + (0.755 + 0.654i)T \)
61 \( 1 + (0.989 - 0.142i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (0.909 + 0.415i)T \)
79 \( 1 + (0.654 - 0.755i)T \)
83 \( 1 + (-0.959 - 0.281i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (0.281 + 0.959i)T \)
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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

−17.75462010415630368786195374659, −17.07873338550968846552962121541, −16.55457476074983288470307731840, −15.59997408016625319729713618562, −15.52407774601461244574401156310, −14.28948440414649977254231482837, −13.805952314014605716760823009089, −13.293320005177847061303519218093, −12.49420763060954120225578770717, −11.75033486723435786431781275652, −11.24431362454579259670714054152, −10.29270200124007674217969908378, −9.79997248499106561045066146927, −9.08200321784433993510622335892, −8.51878214603896583002897579768, −7.40595846966479012266295025112, −6.80271126597920711741923466036, −6.545185465496064984172900261489, −5.409850873305253732199371209254, −4.61414390502236041196685268456, −3.94792263943546081091382728357, −3.30839644950361346820314554446, −2.321200734916277882148028565597, −1.515575247559697368744825540316, −0.52296419213273376273090568428, 0.7889609316926997291892456250, 1.64863196393922207498375560532, 2.75554552158561388177084441428, 3.30042422374949949165759364448, 3.94073745218129432520258392048, 5.05240249844258844013096804378, 5.68449277608055914494924010067, 6.42499490819578973417308087289, 6.80464670009425374727482661622, 8.077126762223985390594314904864, 8.468015695930852207968000421078, 9.17038117834496948364639433599, 9.890203750002914906913136126326, 10.59396733481217267508196480990, 11.3315387030041533773751284880, 11.97187445630477234171612934281, 12.66452963383583985807577421678, 13.31788901364651218671961114043, 13.903526356307243399248433616161, 14.65207935536791755345106306722, 15.48954578000099974061671724696, 16.00098392641911313404945458449, 16.36572529954069867177227454094, 17.487877306093524637327585792884, 17.82945223756554277227200762296

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