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Properties

Label 1-23e2-529.174-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.831 + 0.555i$
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  + (−0.514 − 0.857i)2-s + (−0.896 − 0.443i)3-s + (−0.471 + 0.882i)4-s + (−0.166 − 0.985i)5-s + (0.0806 + 0.996i)6-s + (−0.358 + 0.933i)7-s + (0.998 − 0.0496i)8-s + (0.606 + 0.794i)9-s + (−0.759 + 0.650i)10-s + (0.130 − 0.991i)11-s + (0.813 − 0.581i)12-s + (−0.907 + 0.421i)13-s + (0.984 − 0.172i)14-s + (−0.287 + 0.957i)15-s + (−0.556 − 0.831i)16-s + (0.988 − 0.148i)17-s + ⋯
L(s)  = 1  + (−0.514 − 0.857i)2-s + (−0.896 − 0.443i)3-s + (−0.471 + 0.882i)4-s + (−0.166 − 0.985i)5-s + (0.0806 + 0.996i)6-s + (−0.358 + 0.933i)7-s + (0.998 − 0.0496i)8-s + (0.606 + 0.794i)9-s + (−0.759 + 0.650i)10-s + (0.130 − 0.991i)11-s + (0.813 − 0.581i)12-s + (−0.907 + 0.421i)13-s + (0.984 − 0.172i)14-s + (−0.287 + 0.957i)15-s + (−0.556 − 0.831i)16-s + (0.988 − 0.148i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.831 + 0.555i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.831 + 0.555i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08928823608 - 0.2943617457i\)
\(L(\frac12)\) \(\approx\) \(-0.08928823608 - 0.2943617457i\)
\(L(1)\) \(\approx\) \(0.3819121297 - 0.3112065852i\)
\(L(1)\) \(\approx\) \(0.3819121297 - 0.3112065852i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.514 - 0.857i)T \)
3 \( 1 + (-0.896 - 0.443i)T \)
5 \( 1 + (-0.166 - 0.985i)T \)
7 \( 1 + (-0.358 + 0.933i)T \)
11 \( 1 + (0.130 - 0.991i)T \)
13 \( 1 + (-0.907 + 0.421i)T \)
17 \( 1 + (0.988 - 0.148i)T \)
19 \( 1 + (0.922 - 0.386i)T \)
29 \( 1 + (-0.999 + 0.0124i)T \)
31 \( 1 + (-0.381 - 0.924i)T \)
37 \( 1 + (0.299 - 0.954i)T \)
41 \( 1 + (0.798 - 0.601i)T \)
43 \( 1 + (-0.982 - 0.185i)T \)
47 \( 1 + (0.682 + 0.730i)T \)
53 \( 1 + (-0.759 - 0.650i)T \)
59 \( 1 + (-0.993 - 0.111i)T \)
61 \( 1 + (0.827 - 0.561i)T \)
67 \( 1 + (-0.404 + 0.914i)T \)
71 \( 1 + (0.997 + 0.0744i)T \)
73 \( 1 + (-0.999 - 0.0124i)T \)
79 \( 1 + (-0.0434 + 0.999i)T \)
83 \( 1 + (-0.952 + 0.305i)T \)
89 \( 1 + (-0.996 + 0.0868i)T \)
97 \( 1 + (-0.709 + 0.704i)T \)
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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

−23.71063939423056991669021506736, −23.15079362237391568455884737432, −22.59611892451323170403301093101, −21.92223559647254109692546103043, −20.44419276782646611350833609826, −19.66343858129127703185396681814, −18.57073467978058580800797811559, −17.94366053284578703794313756781, −17.08930518441412032042958028040, −16.55783516638055374454318936095, −15.53434643284335587039168245963, −14.82938216483923141725019622209, −14.13050488805092426089971114015, −12.81528515877697224983101712100, −11.72389447700285719225372911159, −10.60948101140371635829172919390, −10.03009498260235149447165689065, −9.54811374405547955557718722225, −7.643701059406740253757122025891, −7.273994743300846278087443808645, −6.39007120260396005167612491941, −5.395066127022257848808015441084, −4.42221638424169986166120178552, −3.32944530798724970897365858292, −1.3609347611866508453453041352, 0.24859295605840455900492814258, 1.39821324185463576695609062373, 2.55074693360439416373147451737, 3.879548430986236425131168406307, 5.13546302329249126232678507294, 5.748581247144146192433936980914, 7.28214785413503895652226543246, 8.10325634189312078999577062375, 9.22570286360376336851091186454, 9.743941729834954127706328620371, 11.18738743390774268405821059542, 11.72150513269708616705904814164, 12.46283492408650061266709407477, 13.02356003493203013210922544818, 14.10418152953866539109975390555, 15.80662244688627751610809735589, 16.528125759011149117682112579503, 17.03348613181242251784910805772, 18.06328193327960868157265776846, 18.929874231776211171767444549809, 19.323246584508862518253250660476, 20.42926236936211355428489015786, 21.43941674080232299957424953161, 21.97824655159030699749265935227, 22.73789276955287115729336235742

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