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Properties

Label 1-4235-4235.1223-r1-0-0
Degree $1$
Conductor $4235$
Sign $0.312 - 0.950i$
Analytic cond. $455.113$
Root an. cond. $455.113$
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  + (−0.508 + 0.861i)2-s + (0.743 − 0.669i)3-s + (−0.483 − 0.875i)4-s + (0.198 + 0.980i)6-s + (0.999 + 0.0285i)8-s + (0.104 − 0.994i)9-s + (−0.945 − 0.327i)12-s + (−0.996 + 0.0855i)13-s + (−0.532 + 0.846i)16-s + (−0.893 − 0.449i)17-s + (0.803 + 0.595i)18-s + (0.710 − 0.703i)19-s + (0.371 + 0.928i)23-s + (0.761 − 0.647i)24-s + (0.432 − 0.901i)26-s + (−0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.508 + 0.861i)2-s + (0.743 − 0.669i)3-s + (−0.483 − 0.875i)4-s + (0.198 + 0.980i)6-s + (0.999 + 0.0285i)8-s + (0.104 − 0.994i)9-s + (−0.945 − 0.327i)12-s + (−0.996 + 0.0855i)13-s + (−0.532 + 0.846i)16-s + (−0.893 − 0.449i)17-s + (0.803 + 0.595i)18-s + (0.710 − 0.703i)19-s + (0.371 + 0.928i)23-s + (0.761 − 0.647i)24-s + (0.432 − 0.901i)26-s + (−0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.312 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.312 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.312 - 0.950i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ 0.312 - 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.504154868 - 1.088950245i\)
\(L(\frac12)\) \(\approx\) \(1.504154868 - 1.088950245i\)
\(L(1)\) \(\approx\) \(0.9947899756 + 0.01932335829i\)
\(L(1)\) \(\approx\) \(0.9947899756 + 0.01932335829i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.508 + 0.861i)T \)
3 \( 1 + (0.743 - 0.669i)T \)
13 \( 1 + (-0.996 + 0.0855i)T \)
17 \( 1 + (-0.893 - 0.449i)T \)
19 \( 1 + (0.710 - 0.703i)T \)
23 \( 1 + (0.371 + 0.928i)T \)
29 \( 1 + (0.774 + 0.633i)T \)
31 \( 1 + (0.290 - 0.956i)T \)
37 \( 1 + (0.992 - 0.123i)T \)
41 \( 1 + (-0.736 - 0.676i)T \)
43 \( 1 + (0.281 + 0.959i)T \)
47 \( 1 + (0.803 - 0.595i)T \)
53 \( 1 + (0.846 - 0.532i)T \)
59 \( 1 + (-0.217 - 0.976i)T \)
61 \( 1 + (0.00951 + 0.999i)T \)
67 \( 1 + (0.814 - 0.580i)T \)
71 \( 1 + (-0.362 + 0.931i)T \)
73 \( 1 + (0.938 + 0.345i)T \)
79 \( 1 + (0.380 - 0.924i)T \)
83 \( 1 + (-0.717 + 0.696i)T \)
89 \( 1 + (0.0475 - 0.998i)T \)
97 \( 1 + (-0.336 + 0.941i)T \)
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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

−18.51173755588936261937398256663, −17.7513432064728860124631183627, −16.95301529101858313723914226922, −16.50168386702197702715801883579, −15.61365953422834257239621247326, −14.98652792448815503994614483448, −14.08911577757254457449332083771, −13.68679036551563944996156992655, −12.73993815090231065869892729927, −12.212040992317336914670201763438, −11.35495960973979909109041170761, −10.5617492416159154723763777607, −10.12308568142026868718207238196, −9.46659205856492770746740453570, −8.77206722517616098170747450101, −8.18814962484599599760476100220, −7.52069403305226045829839595963, −6.66602733489607564402766228048, −5.36790708021362245959837687876, −4.5472454001297398641132323404, −4.1059054880603203000767804996, −3.08716797739369700712330493228, −2.590468997945436388809822635206, −1.83971607433455227663943117092, −0.77366951821428046777959423507, 0.38454031665072269160807533490, 1.07304250472374798161596622831, 2.1314883706843374097971822615, 2.76308673582051347699856495144, 3.90479710416010800113301913503, 4.79018964224586584276829360409, 5.490048302596189248899867973755, 6.50472247601158095589701783513, 7.03461653805365873227719492348, 7.56239940323194751966797762063, 8.24960401755660063494820247445, 9.107241651048719957402669160188, 9.43655126195269798968530385427, 10.19114987824693246471122853727, 11.26916233270215873418585601163, 11.905760864587488532578298313478, 12.96480742581032716492840651605, 13.44925615301503128101923410918, 14.09228926969983242828936892676, 14.72815724344948832631489821767, 15.39978455128957085040243556450, 15.860016577343539914241353096582, 16.87186325960959258906677327862, 17.48860917378838410271172525148, 18.021886529325049934523716223603

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