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Properties

Label 1-23e2-529.93-r0-0-0
Degree $1$
Conductor $529$
Sign $0.0267 - 0.999i$
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  + (0.854 − 0.519i)2-s + (0.854 − 0.519i)3-s + (0.460 − 0.887i)4-s + (0.682 − 0.730i)5-s + (0.460 − 0.887i)6-s + (0.460 + 0.887i)7-s + (−0.0682 − 0.997i)8-s + (0.460 − 0.887i)9-s + (0.203 − 0.979i)10-s + (−0.917 + 0.398i)11-s + (−0.0682 − 0.997i)12-s + (0.203 + 0.979i)13-s + (0.854 + 0.519i)14-s + (0.203 − 0.979i)15-s + (−0.576 − 0.816i)16-s + (0.203 + 0.979i)17-s + ⋯
L(s)  = 1  + (0.854 − 0.519i)2-s + (0.854 − 0.519i)3-s + (0.460 − 0.887i)4-s + (0.682 − 0.730i)5-s + (0.460 − 0.887i)6-s + (0.460 + 0.887i)7-s + (−0.0682 − 0.997i)8-s + (0.460 − 0.887i)9-s + (0.203 − 0.979i)10-s + (−0.917 + 0.398i)11-s + (−0.0682 − 0.997i)12-s + (0.203 + 0.979i)13-s + (0.854 + 0.519i)14-s + (0.203 − 0.979i)15-s + (−0.576 − 0.816i)16-s + (0.203 + 0.979i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0267 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0267 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.0267 - 0.999i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.0267 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.436725746 - 2.372460793i\)
\(L(\frac12)\) \(\approx\) \(2.436725746 - 2.372460793i\)
\(L(1)\) \(\approx\) \(2.058378828 - 1.232531262i\)
\(L(1)\) \(\approx\) \(2.058378828 - 1.232531262i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.854 - 0.519i)T \)
3 \( 1 + (0.854 - 0.519i)T \)
5 \( 1 + (0.682 - 0.730i)T \)
7 \( 1 + (0.460 + 0.887i)T \)
11 \( 1 + (-0.917 + 0.398i)T \)
13 \( 1 + (0.203 + 0.979i)T \)
17 \( 1 + (0.203 + 0.979i)T \)
19 \( 1 + (0.854 - 0.519i)T \)
29 \( 1 + (-0.917 + 0.398i)T \)
31 \( 1 + (-0.334 - 0.942i)T \)
37 \( 1 + (-0.576 + 0.816i)T \)
41 \( 1 + (-0.775 - 0.631i)T \)
43 \( 1 + (-0.990 + 0.136i)T \)
47 \( 1 + (-0.334 + 0.942i)T \)
53 \( 1 + (0.203 + 0.979i)T \)
59 \( 1 + (0.854 + 0.519i)T \)
61 \( 1 + (0.682 - 0.730i)T \)
67 \( 1 + (-0.917 + 0.398i)T \)
71 \( 1 + (-0.775 + 0.631i)T \)
73 \( 1 + (-0.917 - 0.398i)T \)
79 \( 1 + (-0.990 + 0.136i)T \)
83 \( 1 + (0.682 - 0.730i)T \)
89 \( 1 + (0.962 + 0.269i)T \)
97 \( 1 + (-0.775 + 0.631i)T \)
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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

−23.606867398282614000340445465, −22.75852337348892718273507911304, −22.09157997991810463117491971494, −21.02137072117587877997690565778, −20.75699961457513474303779681326, −19.87741133552135329389812926191, −18.444835374279036080787510945235, −17.77450556164692350043974902875, −16.575820970700036716999586022651, −15.91427727209763054814351730955, −14.91669444069636253266840716984, −14.338917743678212680706093572153, −13.49887212486071355114820827460, −13.18855086337826993428275687857, −11.529310504472489955476253777370, −10.5826623422253961269906495019, −9.95397048754076950772744090545, −8.528033720926291480017926873496, −7.63504612817493877252833875172, −7.06720561267768479446832925596, −5.5489395832746548424883592221, −5.01015332218591715362507692210, −3.54415383607337656443814032601, −3.11168560643580437703686237628, −1.927193693390384149006936372337, 1.51981612622276262441135303735, 2.016564677173767735810520968372, 3.039329640455404342591491784718, 4.324084363876921716375821338735, 5.287490078218007907029330744, 6.11867000893369623629011296063, 7.30663741519985858328987260443, 8.51162341973952079183155132691, 9.2997982249296641990099712457, 10.13731903156839980158748331704, 11.511703909787284672583457067695, 12.3008885643840075644677697697, 13.08787445011779077711256748857, 13.62212402762519509565782817866, 14.60644444052136075495550426941, 15.26074702200409230234487501308, 16.213096747856018401171406539, 17.60796797755512156620482796249, 18.55000666923598282212471282595, 19.07489964192017920891930351617, 20.360727194762303099106918647603, 20.66103044204421966966496158921, 21.50450496499371062863595982356, 22.096787630202426067786560821724, 23.6577155644187878153965820564

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