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Properties

Label 1-547-547.118-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.126 - 0.991i$
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  + (0.924 − 0.381i)2-s + (−0.222 + 0.974i)3-s + (0.709 − 0.705i)4-s + (0.605 − 0.795i)5-s + (0.166 + 0.986i)6-s + (−0.857 − 0.514i)7-s + (0.386 − 0.922i)8-s + (−0.900 − 0.433i)9-s + (0.256 − 0.966i)10-s + (−0.632 − 0.774i)11-s + (0.529 + 0.848i)12-s + (0.365 + 0.930i)13-s + (−0.988 − 0.149i)14-s + (0.641 + 0.767i)15-s + (0.00575 − 0.999i)16-s + (−0.819 − 0.572i)17-s + ⋯
L(s)  = 1  + (0.924 − 0.381i)2-s + (−0.222 + 0.974i)3-s + (0.709 − 0.705i)4-s + (0.605 − 0.795i)5-s + (0.166 + 0.986i)6-s + (−0.857 − 0.514i)7-s + (0.386 − 0.922i)8-s + (−0.900 − 0.433i)9-s + (0.256 − 0.966i)10-s + (−0.632 − 0.774i)11-s + (0.529 + 0.848i)12-s + (0.365 + 0.930i)13-s + (−0.988 − 0.149i)14-s + (0.641 + 0.767i)15-s + (0.00575 − 0.999i)16-s + (−0.819 − 0.572i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.126 - 0.991i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.126 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.217679656 - 1.382545023i\)
\(L(\frac12)\) \(\approx\) \(1.217679656 - 1.382545023i\)
\(L(1)\) \(\approx\) \(1.418266175 - 0.5279204730i\)
\(L(1)\) \(\approx\) \(1.418266175 - 0.5279204730i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.924 - 0.381i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (0.605 - 0.795i)T \)
7 \( 1 + (-0.857 - 0.514i)T \)
11 \( 1 + (-0.632 - 0.774i)T \)
13 \( 1 + (0.365 + 0.930i)T \)
17 \( 1 + (-0.819 - 0.572i)T \)
19 \( 1 + (-0.439 - 0.898i)T \)
23 \( 1 + (0.0287 - 0.999i)T \)
29 \( 1 + (0.322 + 0.946i)T \)
31 \( 1 + (0.675 + 0.736i)T \)
37 \( 1 + (-0.459 - 0.888i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.983 - 0.183i)T \)
47 \( 1 + (-0.919 + 0.391i)T \)
53 \( 1 + (0.343 - 0.939i)T \)
59 \( 1 + (0.428 + 0.903i)T \)
61 \( 1 + (-0.667 + 0.744i)T \)
67 \( 1 + (0.469 - 0.882i)T \)
71 \( 1 + (0.932 + 0.359i)T \)
73 \( 1 + (0.641 - 0.767i)T \)
79 \( 1 + (-0.650 - 0.759i)T \)
83 \( 1 + (0.948 - 0.316i)T \)
89 \( 1 + (0.978 - 0.205i)T \)
97 \( 1 + (0.211 + 0.977i)T \)
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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

−23.3491597952983610821707930836, −22.8181561141175775268928237891, −22.31839399284198245083398776775, −21.338618851644278078239719253689, −20.36324674217950362381622795396, −19.34710673205831889340900465145, −18.518314766552226569702774640816, −17.585648833347699429323459124610, −17.074536847201052460640812619269, −15.56638042896907023227478558515, −15.239778263068047675129837616914, −14.03228549341147932160204690108, −13.24130260422771494770602361429, −12.79918520041276151481316222691, −11.88917731484667904464251671988, −10.8486322526049872859684493161, −9.93200105468570844995425982819, −8.36395850545656176425177007871, −7.53324521256276225466370655493, −6.53681962129293616469141476576, −6.05185079444472396943014124358, −5.25334672977973079262448468633, −3.62459838898699212965758788123, −2.61519576111851068629586020891, −1.95849841360031054934552911412, 0.67494569748095587762267054061, 2.38563578342712895387117378015, 3.352640833950950588550575817764, 4.46262123004628346794268064525, 4.98133713822426313509267153889, 6.130832308196210839599222579281, 6.743734301189211826562387540509, 8.701297623660197526834478782107, 9.36835291842058495557758957624, 10.41306927899196576850561977903, 10.950730944685350167828180491115, 12.03246263060026615708704653798, 13.05454802220507723441441873708, 13.66699591998192589743739939735, 14.44394055248510662214330992291, 15.83725503111171037653175066979, 16.110151364555162585819289395021, 16.83360197810840712089806370344, 18.07426326862288654494466298967, 19.432599713528988413404362802342, 20.04308463402788118885397086964, 20.99047166816003499442470464312, 21.38880831117017960738807036766, 22.19527795577384561652073457486, 22.99834720169362807688968893506

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