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Properties

Label 1-4025-4025.923-r0-0-0
Degree $1$
Conductor $4025$
Sign $0.789 + 0.613i$
Analytic cond. $18.6920$
Root an. cond. $18.6920$
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  + (0.999 + 0.0285i)2-s + (0.996 − 0.0855i)3-s + (0.998 + 0.0570i)4-s + (0.998 − 0.0570i)6-s + (0.996 + 0.0855i)8-s + (0.985 − 0.170i)9-s + (−0.564 + 0.825i)11-s + (0.999 − 0.0285i)12-s + (0.676 + 0.736i)13-s + (0.993 + 0.113i)16-s + (−0.0570 − 0.998i)17-s + (0.989 − 0.142i)18-s + (−0.362 + 0.931i)19-s + (−0.587 + 0.809i)22-s + 24-s + ⋯
L(s)  = 1  + (0.999 + 0.0285i)2-s + (0.996 − 0.0855i)3-s + (0.998 + 0.0570i)4-s + (0.998 − 0.0570i)6-s + (0.996 + 0.0855i)8-s + (0.985 − 0.170i)9-s + (−0.564 + 0.825i)11-s + (0.999 − 0.0285i)12-s + (0.676 + 0.736i)13-s + (0.993 + 0.113i)16-s + (−0.0570 − 0.998i)17-s + (0.989 − 0.142i)18-s + (−0.362 + 0.931i)19-s + (−0.587 + 0.809i)22-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $0.789 + 0.613i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (923, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ 0.789 + 0.613i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.318525456 + 1.821982761i\)
\(L(\frac12)\) \(\approx\) \(5.318525456 + 1.821982761i\)
\(L(1)\) \(\approx\) \(2.845606674 + 0.3728180434i\)
\(L(1)\) \(\approx\) \(2.845606674 + 0.3728180434i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.999 + 0.0285i)T \)
3 \( 1 + (0.996 - 0.0855i)T \)
11 \( 1 + (-0.564 + 0.825i)T \)
13 \( 1 + (0.676 + 0.736i)T \)
17 \( 1 + (-0.0570 - 0.998i)T \)
19 \( 1 + (-0.362 + 0.931i)T \)
29 \( 1 + (0.362 + 0.931i)T \)
31 \( 1 + (-0.0855 + 0.996i)T \)
37 \( 1 + (0.170 + 0.985i)T \)
41 \( 1 + (-0.897 + 0.441i)T \)
43 \( 1 + (0.755 - 0.654i)T \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (-0.491 - 0.870i)T \)
59 \( 1 + (-0.736 + 0.676i)T \)
61 \( 1 + (-0.516 - 0.856i)T \)
67 \( 1 + (-0.791 - 0.610i)T \)
71 \( 1 + (0.941 + 0.336i)T \)
73 \( 1 + (-0.967 + 0.254i)T \)
79 \( 1 + (-0.198 + 0.980i)T \)
83 \( 1 + (-0.884 + 0.466i)T \)
89 \( 1 + (-0.921 + 0.389i)T \)
97 \( 1 + (-0.884 - 0.466i)T \)
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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

−18.72302578775812518901924659021, −17.691318747899847431595626330341, −16.86337235758699755337209093624, −15.94175558014368401207217138433, −15.51737171977422467118589690278, −15.038111835593041339136652682081, −14.20040562409400198115727957687, −13.607371961564911553731207572514, −13.05146998509296556611822385286, −12.64935179603146127492052165495, −11.55455320242200709769631315027, −10.73016866890695339066444407432, −10.43527416811519870064044356680, −9.32529217026470845292138669218, −8.49833332080067137137402497200, −7.88539169890428035879511807028, −7.28451692649449982389127419371, −6.13257965537077791712720538270, −5.83228866652855458577988391649, −4.64979062360739670210764971527, −4.09229011765814830430995869613, −3.27628323007241497115307938287, −2.72157959467143071693482028185, −1.961137161545375202306206364662, −0.91356626661608414030614108834, 1.41186896726365122043928385563, 1.92246639590217154898972379088, 2.862099746174502536904474240796, 3.41354601409328956220360655712, 4.3221432674936746400771525446, 4.81968110978167783226464438123, 5.7533299634310781108958846683, 6.908274273038868946108080135365, 7.00501853212742439569827146202, 8.05431588381975347858808086144, 8.633568455982241818612253655911, 9.63748613163868611636503555225, 10.27211550925924665581874940922, 11.02436939801656154515381280809, 12.01194961464801745596723536459, 12.4921263307451538381365574851, 13.18711613429945409857171211301, 14.00854625654682972208171589895, 14.15047670213294713935651912391, 15.135195316134011022346851230952, 15.56591920514422807803973797347, 16.25096709000727855103475948077, 16.90501538905035793054261432492, 18.17690193506256320191500041283, 18.5316496220110320409252807540

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