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Properties

Label 1-23e2-529.197-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.707 - 0.706i$
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  + (−0.404 + 0.914i)2-s + (0.154 + 0.987i)3-s + (−0.673 − 0.739i)4-s + (0.299 + 0.954i)5-s + (−0.966 − 0.257i)6-s + (0.999 + 0.0248i)7-s + (0.948 − 0.317i)8-s + (−0.952 + 0.305i)9-s + (−0.993 − 0.111i)10-s + (−0.998 − 0.0620i)11-s + (0.626 − 0.779i)12-s + (−0.311 + 0.950i)13-s + (−0.426 + 0.904i)14-s + (−0.896 + 0.443i)15-s + (−0.0929 + 0.995i)16-s + (0.566 − 0.824i)17-s + ⋯
L(s)  = 1  + (−0.404 + 0.914i)2-s + (0.154 + 0.987i)3-s + (−0.673 − 0.739i)4-s + (0.299 + 0.954i)5-s + (−0.966 − 0.257i)6-s + (0.999 + 0.0248i)7-s + (0.948 − 0.317i)8-s + (−0.952 + 0.305i)9-s + (−0.993 − 0.111i)10-s + (−0.998 − 0.0620i)11-s + (0.626 − 0.779i)12-s + (−0.311 + 0.950i)13-s + (−0.426 + 0.904i)14-s + (−0.896 + 0.443i)15-s + (−0.0929 + 0.995i)16-s + (0.566 − 0.824i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.707 - 0.706i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.707 - 0.706i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3122400773 + 0.7548225869i\)
\(L(\frac12)\) \(\approx\) \(-0.3122400773 + 0.7548225869i\)
\(L(1)\) \(\approx\) \(0.4354660387 + 0.6887335037i\)
\(L(1)\) \(\approx\) \(0.4354660387 + 0.6887335037i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.404 + 0.914i)T \)
3 \( 1 + (0.154 + 0.987i)T \)
5 \( 1 + (0.299 + 0.954i)T \)
7 \( 1 + (0.999 + 0.0248i)T \)
11 \( 1 + (-0.998 - 0.0620i)T \)
13 \( 1 + (-0.311 + 0.950i)T \)
17 \( 1 + (0.566 - 0.824i)T \)
19 \( 1 + (-0.847 - 0.530i)T \)
29 \( 1 + (0.0806 + 0.996i)T \)
31 \( 1 + (-0.982 + 0.185i)T \)
37 \( 1 + (0.369 + 0.929i)T \)
41 \( 1 + (-0.492 + 0.870i)T \)
43 \( 1 + (-0.935 + 0.352i)T \)
47 \( 1 + (-0.576 + 0.816i)T \)
53 \( 1 + (-0.993 + 0.111i)T \)
59 \( 1 + (0.664 - 0.747i)T \)
61 \( 1 + (-0.743 + 0.668i)T \)
67 \( 1 + (0.940 + 0.340i)T \)
71 \( 1 + (-0.885 - 0.465i)T \)
73 \( 1 + (0.0806 - 0.996i)T \)
79 \( 1 + (0.481 + 0.876i)T \)
83 \( 1 + (0.901 - 0.432i)T \)
89 \( 1 + (-0.535 - 0.844i)T \)
97 \( 1 + (0.931 - 0.363i)T \)
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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

−23.20861842331588746447951825218, −21.80270125913001477911212766186, −20.9522371938678591415594123732, −20.47775293510368152644454354721, −19.6587098699195738552901293893, −18.74135024875895095103373634300, −17.963537458186730618082984712785, −17.338935642060985690337737447142, −16.690278209884233700340673194461, −15.10618921627626536162980002341, −14.07708765162427119603250746999, −13.09141335523876598169438864771, −12.682450444470867498424007799627, −11.89430769354864473711277601809, −10.83068686309145053715366651683, −9.9850720478577044513117325566, −8.68610964186971719710888144422, −8.12201654290209829010900555162, −7.5775578107344105905993893609, −5.758891962073622143352043663032, −5.0080038814176182034313083494, −3.70188674581303896218151742578, −2.27514913613650446518267124930, −1.70773600922432487008293470792, −0.4697674073632039529943522049, 1.91998328368704572994773909644, 3.169204013065580975421289107384, 4.64488111924525570812718656195, 5.12030385663555742912061054258, 6.28561155053467691601622638333, 7.34310072180928761314912957233, 8.18853694443572476059958398105, 9.16468845184571137102221432538, 10.00530127863903853689948995896, 10.79342947384410398580541593037, 11.471259642985461083365678506156, 13.321037439827666097664999034236, 14.314503333989969499813466629675, 14.635814662815205913132174008542, 15.452261184937748913501120186745, 16.35835131047534990471991120265, 17.12178757520518985296108286825, 18.09027005337030083572343745651, 18.62178286776472070176182876210, 19.70154425161704813724529755925, 20.84536505496474447360643764454, 21.62689112716604738747744334525, 22.24917557611681459302103972369, 23.440418348443515594075046834155, 23.7995126836383397185602777202

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