Properties

Label 1-23e2-529.312-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.865 - 0.500i$
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  + (−0.885 − 0.465i)2-s + (−0.996 + 0.0868i)3-s + (0.566 + 0.824i)4-s + (−0.834 − 0.551i)5-s + (0.922 + 0.386i)6-s + (−0.993 + 0.111i)7-s + (−0.117 − 0.993i)8-s + (0.984 − 0.172i)9-s + (0.481 + 0.876i)10-s + (0.275 + 0.961i)11-s + (−0.635 − 0.771i)12-s + (−0.596 − 0.802i)13-s + (0.931 + 0.363i)14-s + (0.879 + 0.476i)15-s + (−0.358 + 0.933i)16-s + (0.346 + 0.938i)17-s + ⋯
L(s)  = 1  + (−0.885 − 0.465i)2-s + (−0.996 + 0.0868i)3-s + (0.566 + 0.824i)4-s + (−0.834 − 0.551i)5-s + (0.922 + 0.386i)6-s + (−0.993 + 0.111i)7-s + (−0.117 − 0.993i)8-s + (0.984 − 0.172i)9-s + (0.481 + 0.876i)10-s + (0.275 + 0.961i)11-s + (−0.635 − 0.771i)12-s + (−0.596 − 0.802i)13-s + (0.931 + 0.363i)14-s + (0.879 + 0.476i)15-s + (−0.358 + 0.933i)16-s + (0.346 + 0.938i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.865 - 0.500i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (312, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.865 - 0.500i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04619562325 - 0.1721874845i\)
\(L(\frac12)\) \(\approx\) \(0.04619562325 - 0.1721874845i\)
\(L(1)\) \(\approx\) \(0.3602170949 - 0.09158014713i\)
\(L(1)\) \(\approx\) \(0.3602170949 - 0.09158014713i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.885 - 0.465i)T \)
3 \( 1 + (-0.996 + 0.0868i)T \)
5 \( 1 + (-0.834 - 0.551i)T \)
7 \( 1 + (-0.993 + 0.111i)T \)
11 \( 1 + (0.275 + 0.961i)T \)
13 \( 1 + (-0.596 - 0.802i)T \)
17 \( 1 + (0.346 + 0.938i)T \)
19 \( 1 + (0.586 - 0.809i)T \)
29 \( 1 + (0.912 - 0.409i)T \)
31 \( 1 + (-0.743 + 0.668i)T \)
37 \( 1 + (0.606 + 0.794i)T \)
41 \( 1 + (-0.999 - 0.0372i)T \)
43 \( 1 + (0.998 + 0.0496i)T \)
47 \( 1 + (-0.917 + 0.398i)T \)
53 \( 1 + (0.481 - 0.876i)T \)
59 \( 1 + (-0.791 - 0.611i)T \)
61 \( 1 + (0.154 - 0.987i)T \)
67 \( 1 + (0.00620 - 0.999i)T \)
71 \( 1 + (-0.820 + 0.571i)T \)
73 \( 1 + (0.912 + 0.409i)T \)
79 \( 1 + (-0.0929 - 0.995i)T \)
83 \( 1 + (-0.426 + 0.904i)T \)
89 \( 1 + (-0.982 - 0.185i)T \)
97 \( 1 + (0.105 - 0.994i)T \)
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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

−23.854051655695150913696558997584, −23.07380027535650857046643807469, −22.46085332258446355056897291705, −21.48842255831155962057867586096, −20.09515005466133984579706677437, −19.26222343489318327075454198212, −18.71931769408885376245199557211, −18.086419868028402729856936014664, −16.7471575734288509550666002527, −16.37993286158344840804675542359, −15.82473548908677234572965919628, −14.66698932739858613620269078331, −13.72304674116484262540997979079, −12.185791548582947119392524913224, −11.65823209173759314020009691879, −10.77236988567546803831329892475, −9.948248166440540205866443789389, −9.083171396318474528531742946006, −7.71001532636885038110949889156, −7.04894617113772069623927876715, −6.318677170134938666766107911189, −5.41730960334156809865681750472, −4.02811205238035904547988926523, −2.750769426220903354932673798938, −1.014509909324409445801895153370, 0.19145828382054755893407406341, 1.40570480004743965874426705134, 3.01716706809577934160163918927, 4.05813151400259812674001054934, 5.12540732725572482139291965012, 6.50024705653535641210421237643, 7.24626747170679408585445744163, 8.18798924427797690658460529316, 9.44514847780220196640281306390, 10.00610975098576636951574891773, 10.95739289599581408252894445444, 11.9601656879434351960137839566, 12.49333212310405910871447123375, 13.02609048819812732131213374260, 15.182355271443886163071497133143, 15.70994246328457723659907816579, 16.54716191829646571062331648679, 17.24589806740542702369018808750, 17.94985649104032927605650664025, 19.013944980146345004650874355434, 19.74127250698324239459131122545, 20.299207795135907597515964790955, 21.50719971174548738050080384598, 22.29210476683746521510921752863, 22.98464550646612183907809649795

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