Properties

Label 1-547-547.505-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.983 - 0.181i$
Analytic cond. $2.54025$
Root an. cond. $2.54025$
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  + (−0.988 + 0.149i)2-s + (−0.222 + 0.974i)3-s + (0.955 − 0.294i)4-s + (0.0747 + 0.997i)5-s + (0.0747 − 0.997i)6-s + (0.955 + 0.294i)7-s + (−0.900 + 0.433i)8-s + (−0.900 − 0.433i)9-s + (−0.222 − 0.974i)10-s + (−0.5 + 0.866i)11-s + (0.0747 + 0.997i)12-s + (0.365 + 0.930i)13-s + (−0.988 − 0.149i)14-s + (−0.988 − 0.149i)15-s + (0.826 − 0.563i)16-s + (0.826 − 0.563i)17-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)2-s + (−0.222 + 0.974i)3-s + (0.955 − 0.294i)4-s + (0.0747 + 0.997i)5-s + (0.0747 − 0.997i)6-s + (0.955 + 0.294i)7-s + (−0.900 + 0.433i)8-s + (−0.900 − 0.433i)9-s + (−0.222 − 0.974i)10-s + (−0.5 + 0.866i)11-s + (0.0747 + 0.997i)12-s + (0.365 + 0.930i)13-s + (−0.988 − 0.149i)14-s + (−0.988 − 0.149i)15-s + (0.826 − 0.563i)16-s + (0.826 − 0.563i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.983 - 0.181i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.983 - 0.181i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05895301461 + 0.6436757668i\)
\(L(\frac12)\) \(\approx\) \(-0.05895301461 + 0.6436757668i\)
\(L(1)\) \(\approx\) \(0.4699995829 + 0.4419142302i\)
\(L(1)\) \(\approx\) \(0.4699995829 + 0.4419142302i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.988 + 0.149i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (0.0747 + 0.997i)T \)
7 \( 1 + (0.955 + 0.294i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.365 + 0.930i)T \)
17 \( 1 + (0.826 - 0.563i)T \)
19 \( 1 + (-0.988 - 0.149i)T \)
23 \( 1 + (-0.988 - 0.149i)T \)
29 \( 1 + (-0.900 + 0.433i)T \)
31 \( 1 + (-0.222 + 0.974i)T \)
37 \( 1 + (0.826 - 0.563i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.955 + 0.294i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.365 + 0.930i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.826 - 0.563i)T \)
67 \( 1 + (0.826 - 0.563i)T \)
71 \( 1 + (0.826 - 0.563i)T \)
73 \( 1 + (-0.988 + 0.149i)T \)
79 \( 1 + (-0.222 - 0.974i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 + (-0.988 - 0.149i)T \)
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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

−23.39987177465889280202083743897, −21.88106511714612690585694320648, −20.80802399848278708853645283938, −20.46509048576433660389290214607, −19.40766471064267649227602649684, −18.700267337048079444990199607449, −17.87104370512416007287808393137, −17.135253446292107488457740053453, −16.65364872268561626166937158817, −15.55824040951793864059305877268, −14.385423681147345440942243130885, −13.20081423504689064274064499177, −12.62290536040221855362590838669, −11.59570135111917927398915985335, −10.95098093690675169775460796979, −9.940455594985785086949138711053, −8.476938453511996284624398213160, −8.19928069559748517728600921291, −7.53376674247286884730034344389, −5.99945087714567957240877085396, −5.53812109498486173966533943922, −3.83900495973619420463295637169, −2.328669350064393077999698543099, −1.42099132900641553457404337304, −0.48434842885372894481847607260, 1.805395937397902211117990465585, 2.71738332474673851969909689136, 4.06020964495959673650134634684, 5.26079852452915079558605740328, 6.21838325906466919814784027141, 7.24905619467935822639213676072, 8.17407466446073096423934377977, 9.194597897346996334345019076337, 9.9900223349546412391446067775, 10.77346652323393231858594180591, 11.36712456503395102433016523649, 12.21069683393694036735137598715, 14.183193592586923083292021967293, 14.71693569549612828410903260802, 15.39962830783444898703676904507, 16.29373379788233381563327875364, 17.120431372945886020572805338, 18.14343669912990995664016790191, 18.35863072725067343483855097870, 19.61010757280799374466723325886, 20.55633050836955055328324266428, 21.27652838856619716047096154890, 21.83241134899438124796650540749, 23.20411285932041975378707338479, 23.62078606083013045110191369369

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