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Properties

Label 1-65e2-4225.303-r1-0-0
Degree $1$
Conductor $4225$
Sign $0.925 - 0.379i$
Analytic cond. $454.039$
Root an. cond. $454.039$
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  + (−0.832 − 0.554i)2-s + (−0.561 − 0.827i)3-s + (0.384 + 0.923i)4-s + (0.00805 + 0.999i)6-s + (0.391 − 0.919i)7-s + (0.192 − 0.981i)8-s + (−0.369 + 0.929i)9-s + (−0.369 − 0.929i)11-s + (0.548 − 0.836i)12-s + (−0.836 + 0.548i)14-s + (−0.704 + 0.709i)16-s + (0.753 + 0.657i)17-s + (0.822 − 0.568i)18-s + (0.669 + 0.743i)19-s + (−0.981 + 0.192i)21-s + (−0.207 + 0.978i)22-s + ⋯
L(s)  = 1  + (−0.832 − 0.554i)2-s + (−0.561 − 0.827i)3-s + (0.384 + 0.923i)4-s + (0.00805 + 0.999i)6-s + (0.391 − 0.919i)7-s + (0.192 − 0.981i)8-s + (−0.369 + 0.929i)9-s + (−0.369 − 0.929i)11-s + (0.548 − 0.836i)12-s + (−0.836 + 0.548i)14-s + (−0.704 + 0.709i)16-s + (0.753 + 0.657i)17-s + (0.822 − 0.568i)18-s + (0.669 + 0.743i)19-s + (−0.981 + 0.192i)21-s + (−0.207 + 0.978i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $0.925 - 0.379i$
Analytic conductor: \(454.039\)
Root analytic conductor: \(454.039\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4225} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4225,\ (1:\ ),\ 0.925 - 0.379i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8007813433 - 0.1576481858i\)
\(L(\frac12)\) \(\approx\) \(0.8007813433 - 0.1576481858i\)
\(L(1)\) \(\approx\) \(0.5300562844 - 0.3082802020i\)
\(L(1)\) \(\approx\) \(0.5300562844 - 0.3082802020i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.832 - 0.554i)T \)
3 \( 1 + (-0.561 - 0.827i)T \)
7 \( 1 + (0.391 - 0.919i)T \)
11 \( 1 + (-0.369 - 0.929i)T \)
17 \( 1 + (0.753 + 0.657i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (-0.994 - 0.104i)T \)
29 \( 1 + (-0.937 + 0.347i)T \)
31 \( 1 + (0.861 - 0.506i)T \)
37 \( 1 + (0.910 - 0.414i)T \)
41 \( 1 + (0.789 + 0.613i)T \)
43 \( 1 + (0.979 - 0.200i)T \)
47 \( 1 + (-0.285 + 0.958i)T \)
53 \( 1 + (-0.421 - 0.906i)T \)
59 \( 1 + (-0.892 + 0.450i)T \)
61 \( 1 + (-0.293 - 0.955i)T \)
67 \( 1 + (-0.923 - 0.384i)T \)
71 \( 1 + (-0.899 - 0.435i)T \)
73 \( 1 + (-0.144 + 0.989i)T \)
79 \( 1 + (-0.958 - 0.285i)T \)
83 \( 1 + (0.997 + 0.0724i)T \)
89 \( 1 + (-0.104 + 0.994i)T \)
97 \( 1 + (0.840 + 0.541i)T \)
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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

−18.106229610441428771026968933033, −17.60755235677733526159131348625, −16.95223281011332769132773246630, −16.06038034868194269234025087721, −15.75136665615132275016749694365, −15.10378592096034184609008772248, −14.556626044050576030570798772069, −13.78276846324944365188717274839, −12.537091647621032087600706141813, −11.77021060843596339316215847307, −11.43659425132976838121295241707, −10.45736530250889467584889217967, −9.904753547715623273606746379495, −9.30126259397002657209746767911, −8.77883489938067577735217499481, −7.72342298207673558055038868579, −7.30574283144056557151438270573, −6.10489172766432715684426167559, −5.75760437402537531113262575156, −4.91304470521325913138280145806, −4.46565764019560158918115542579, −3.0596882475689325479729808562, −2.31301121434947881991948654779, −1.27563764167867238932373072305, −0.26735640797496331371256429976, 0.64729755777545782645965613173, 1.223379196011598088408025729244, 1.95841997210204836347913180832, 2.96071944726287714881998628324, 3.74502370960248197221282231904, 4.61684279118242746854542537792, 5.843977709119448574590905924318, 6.215120547969458581940571315189, 7.35223950578746812354190116212, 7.91726201116588344319347536093, 8.06683060298863148881368458440, 9.259367927508956923277496606434, 10.12560528507433081187844930705, 10.68633061306265530920767334517, 11.26954519700535912966472721359, 11.848620500560990166309276329783, 12.636131610655465475765260713715, 13.19381922308670292031538856811, 13.94062355837352261943303009923, 14.55480857581185158098930295402, 15.96461569229889539512102085890, 16.39848694920468825438361595935, 16.959578188788422205524375085336, 17.584084618228996223388346810816, 18.22769295030944015958733381013

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