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Properties

Label 1-547-547.51-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: $\chi_{547} (51, \cdot )$
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

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

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