Properties

Label 1-5520-5520.3587-r0-0-0
Degree $1$
Conductor $5520$
Sign $-0.811 + 0.584i$
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  i·7-s + i·11-s + 13-s + i·17-s i·19-s + i·29-s − 31-s − 37-s + 41-s − 43-s + i·47-s − 49-s − 53-s + i·59-s i·61-s + ⋯
L(s)  = 1  i·7-s + i·11-s + 13-s + i·17-s i·19-s + i·29-s − 31-s − 37-s + 41-s − 43-s + i·47-s − 49-s − 53-s + i·59-s i·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.811 + 0.584i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (3587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ -0.811 + 0.584i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1526703161 + 0.4729229574i\)
\(L(\frac12)\) \(\approx\) \(0.1526703161 + 0.4729229574i\)
\(L(1)\) \(\approx\) \(0.9216911854 + 0.02975232688i\)
\(L(1)\) \(\approx\) \(0.9216911854 + 0.02975232688i\)

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 \( 1 - iT \)
29 \( 1 \)
31 \( 1 \)
37 \( 1 + iT \)
41 \( 1 \)
43 \( 1 + T \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 + iT \)
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

−17.804974531319029473095979295621, −16.785899561920601926707109584527, −16.169261118406854470259935966488, −15.80179125314167243752546992103, −14.97762350010487516150237407288, −14.2903961324119080397172474665, −13.628656089441319559750425733676, −13.04790307153253919239745461726, −12.16906551542151637809846357863, −11.62668217714862639349334632831, −11.067057957278805880077355741774, −10.25850532609487664508800164927, −9.43531479892860521844988899172, −8.79372777450024821028669014568, −8.30053543959579487471069037117, −7.532352288089759650829248977750, −6.56820481804746686108390753357, −5.840452009662714393493163512128, −5.51633191740723009456631032314, −4.52395244936228736257736466823, −3.5432950604072625442537791900, −3.07674433889435803137291482260, −2.10025661798769147357503756549, −1.333485178703220763041828585356, −0.12362308681782777933663966584, 1.269990797332645708555114986848, 1.70194815178655527961273898247, 2.89241017189975538820499266418, 3.67936772808897255243643274376, 4.30314529758293811571727342782, 4.98082504211718636762696486996, 5.91309214069186043750337679836, 6.71262207177527227339719747613, 7.2197694095647936136223384464, 7.93637637967114250076332213843, 8.770275357617909616680468241888, 9.3761044219663728438327501676, 10.27599539967072993120370052828, 10.753393012978164208714570030787, 11.2982724977707968695010136198, 12.2757504690672707214412103676, 13.00256960046446712796661997610, 13.312142706316873484955924550852, 14.31050597191669764405929109868, 14.6875799417906825032398302712, 15.63077468041964407216555149242, 16.06535697483515489148804050181, 16.94166066473234042469579002049, 17.47647773343211248125866721710, 17.97792422182629172656009516232

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