Properties

Label 1-23e2-529.188-r0-0-0
Degree $1$
Conductor $529$
Sign $0.0956 + 0.995i$
Analytic cond. $2.45666$
Root an. cond. $2.45666$
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  + (0.931 + 0.363i)2-s + (−0.791 − 0.611i)3-s + (0.735 + 0.678i)4-s + (0.154 + 0.987i)5-s + (−0.514 − 0.857i)6-s + (0.922 + 0.386i)7-s + (0.437 + 0.899i)8-s + (0.251 + 0.967i)9-s + (−0.215 + 0.976i)10-s + (0.545 + 0.837i)11-s + (−0.166 − 0.985i)12-s + (0.346 − 0.938i)13-s + (0.717 + 0.696i)14-s + (0.481 − 0.876i)15-s + (0.0806 + 0.996i)16-s + (−0.977 − 0.209i)17-s + ⋯
L(s)  = 1  + (0.931 + 0.363i)2-s + (−0.791 − 0.611i)3-s + (0.735 + 0.678i)4-s + (0.154 + 0.987i)5-s + (−0.514 − 0.857i)6-s + (0.922 + 0.386i)7-s + (0.437 + 0.899i)8-s + (0.251 + 0.967i)9-s + (−0.215 + 0.976i)10-s + (0.545 + 0.837i)11-s + (−0.166 − 0.985i)12-s + (0.346 − 0.938i)13-s + (0.717 + 0.696i)14-s + (0.481 − 0.876i)15-s + (0.0806 + 0.996i)16-s + (−0.977 − 0.209i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0956 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0956 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.0956 + 0.995i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.0956 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.556541483 + 1.414094940i\)
\(L(\frac12)\) \(\approx\) \(1.556541483 + 1.414094940i\)
\(L(1)\) \(\approx\) \(1.479929051 + 0.6203308014i\)
\(L(1)\) \(\approx\) \(1.479929051 + 0.6203308014i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.931 + 0.363i)T \)
3 \( 1 + (-0.791 - 0.611i)T \)
5 \( 1 + (0.154 + 0.987i)T \)
7 \( 1 + (0.922 + 0.386i)T \)
11 \( 1 + (0.545 + 0.837i)T \)
13 \( 1 + (0.346 - 0.938i)T \)
17 \( 1 + (-0.977 - 0.209i)T \)
19 \( 1 + (-0.885 + 0.465i)T \)
29 \( 1 + (0.275 - 0.961i)T \)
31 \( 1 + (-0.986 - 0.160i)T \)
37 \( 1 + (0.975 + 0.221i)T \)
41 \( 1 + (-0.381 + 0.924i)T \)
43 \( 1 + (0.867 + 0.498i)T \)
47 \( 1 + (-0.917 - 0.398i)T \)
53 \( 1 + (-0.215 - 0.976i)T \)
59 \( 1 + (0.586 - 0.809i)T \)
61 \( 1 + (0.664 + 0.747i)T \)
67 \( 1 + (0.751 - 0.659i)T \)
71 \( 1 + (0.105 + 0.994i)T \)
73 \( 1 + (0.275 + 0.961i)T \)
79 \( 1 + (-0.191 - 0.981i)T \)
83 \( 1 + (0.645 - 0.763i)T \)
89 \( 1 + (-0.926 - 0.375i)T \)
97 \( 1 + (0.948 + 0.317i)T \)
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

−23.54019247518826201335734977901, −22.171016079908531140873237835824, −21.61876732738286461409298165592, −21.03865529986494248839634203132, −20.26479990656383518234269259270, −19.41274773064252031372775970207, −18.09465151388152351867213807463, −17.05858210242681541838880441721, −16.47717885251169651335579205725, −15.67881300271382710938648652559, −14.64313772614526538115091526869, −13.83762127615688986562981442313, −12.89402263326956377464771867695, −12.00590268268517165259882047758, −11.12000664080565016665259928899, −10.805505606832213624999071707664, −9.36313369693574302599456740769, −8.62573546949600334662968396863, −6.93967906507238236422986012928, −6.06375883723382205287678333499, −5.14039327935677912504979427076, −4.36460721965903734395979734959, −3.821828394472526258392698651412, −1.96075137524589937816421418937, −0.94417676497474372960914473177, 1.79666798793287181866934785606, 2.50864553814832544130689605299, 4.03521508080866504008646691812, 4.98059262087156069707755950537, 5.99176989583387125169156904637, 6.6003829554704846458268283138, 7.53397145909945503196148423273, 8.2943119841056815040697951414, 10.11198008244985363964590262627, 11.23008198157966294362071710046, 11.461641151753227157545960439322, 12.646389074245462642153402218602, 13.28368094353137774612455937634, 14.434186112147963884822724231025, 14.95927375191656883500511802476, 15.82897725628864847806900506281, 17.09277334419562821240950819303, 17.721524698215896662254619665277, 18.27525016231099704140446321277, 19.50496099932325254681892094740, 20.515974063630940632640699284683, 21.57184709531391623274121235164, 22.20119191019645292661219171496, 22.918787986207758149165500352421, 23.43469893707937738989414092971

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