Properties

Label 1-23e2-529.121-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

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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: $\chi_{529} (121, \cdot )$
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 \( 1 + (0.948 - 0.317i)T \)
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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

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

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