↑

Properties

Label 1-517-517.145-r1-0-0
Degree $1$
Conductor $517$
Sign $-0.462 + 0.886i$
Analytic cond. $55.5593$
Root an. cond. $55.5593$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.385 + 0.922i)2-s + (−0.955 + 0.295i)3-s + (−0.702 + 0.711i)4-s + (0.410 + 0.911i)5-s + (−0.641 − 0.767i)6-s + (−0.0409 − 0.999i)7-s + (−0.927 − 0.373i)8-s + (0.824 − 0.565i)9-s + (−0.682 + 0.730i)10-s + (0.460 − 0.887i)12-s + (−0.256 − 0.966i)13-s + (0.905 − 0.423i)14-s + (−0.662 − 0.749i)15-s + (−0.0136 − 0.999i)16-s + (−0.881 + 0.472i)17-s + (0.839 + 0.542i)18-s + ⋯
L(s)  = 1  + (0.385 + 0.922i)2-s + (−0.955 + 0.295i)3-s + (−0.702 + 0.711i)4-s + (0.410 + 0.911i)5-s + (−0.641 − 0.767i)6-s + (−0.0409 − 0.999i)7-s + (−0.927 − 0.373i)8-s + (0.824 − 0.565i)9-s + (−0.682 + 0.730i)10-s + (0.460 − 0.887i)12-s + (−0.256 − 0.966i)13-s + (0.905 − 0.423i)14-s + (−0.662 − 0.749i)15-s + (−0.0136 − 0.999i)16-s + (−0.881 + 0.472i)17-s + (0.839 + 0.542i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $-0.462 + 0.886i$
Analytic conductor: \(55.5593\)
Root analytic conductor: \(55.5593\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (1:\ ),\ -0.462 + 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7220160632 + 1.190441454i\)
\(L(\frac12)\) \(\approx\) \(0.7220160632 + 1.190441454i\)
\(L(1)\) \(\approx\) \(0.7382151995 + 0.5430779791i\)
\(L(1)\) \(\approx\) \(0.7382151995 + 0.5430779791i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.385 + 0.922i)T \)
3 \( 1 + (-0.955 + 0.295i)T \)
5 \( 1 + (0.410 + 0.911i)T \)
7 \( 1 + (-0.0409 - 0.999i)T \)
13 \( 1 + (-0.256 - 0.966i)T \)
17 \( 1 + (-0.881 + 0.472i)T \)
19 \( 1 + (-0.994 - 0.109i)T \)
23 \( 1 + (0.854 - 0.519i)T \)
29 \( 1 + (0.839 + 0.542i)T \)
31 \( 1 + (0.946 - 0.321i)T \)
37 \( 1 + (0.905 + 0.423i)T \)
41 \( 1 + (0.969 + 0.243i)T \)
43 \( 1 + (-0.460 - 0.887i)T \)
53 \( 1 + (-0.531 + 0.847i)T \)
59 \( 1 + (0.986 + 0.163i)T \)
61 \( 1 + (0.981 + 0.190i)T \)
67 \( 1 + (0.962 - 0.269i)T \)
71 \( 1 + (-0.385 + 0.922i)T \)
73 \( 1 + (-0.792 + 0.609i)T \)
79 \( 1 + (-0.976 - 0.216i)T \)
83 \( 1 + (-0.721 + 0.692i)T \)
89 \( 1 + (0.203 - 0.979i)T \)
97 \( 1 + (-0.0136 + 0.999i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.01785948547526461921831650416, −22.02410066663009354778347161893, −21.42049057169267636389942234271, −20.93010834791618724256775508696, −19.49329997915455874043836971943, −19.06765698137145785100384274454, −17.93003100451385183635200786131, −17.441373583807337003689790792732, −16.31518738806434632464063526511, −15.44038864631264557086480974369, −14.21730633249025892455262058218, −13.12269055168465843275253897773, −12.71329018042034421679212113390, −11.74003184092326448635601933320, −11.30773631009771299295058350962, −10.00922952158337976492524584219, −9.24621869589425215461837442739, −8.391536791216897274299988996446, −6.60867132987012953751198080714, −5.86468575375197203130826027956, −4.86986964215058041997661822168, −4.37839193008833821417006985983, −2.508998544898016109183668746870, −1.730440169400549079660075071692, −0.54627388997621398276965572198, 0.71924783728901327290894467710, 2.78874430186272731100573459537, 4.00981019087473990128102793365, 4.77255618391822370921989566684, 5.92727709244549399646534374689, 6.634851834842273710455763215412, 7.24472559027292998945664181426, 8.46759199075130774494280448861, 9.865265082763776336712164532616, 10.52202289642500857550705667960, 11.35006447605737433211348538993, 12.749348330093839982285081805676, 13.24821202379283522644275294994, 14.41301109870945277718621500447, 15.12878039789204551678857259799, 15.89252422572940720773166707274, 17.08721322571854102634159605797, 17.33778935224916849304263821835, 18.114292676939844233488906033, 19.1505569732632897500127743526, 20.53121782993281797746597289566, 21.56731976358343631191523538006, 22.12273362575747192502390052474, 22.9930039923529192101811874694, 23.31468973370438274958174437465

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