↑

Properties

Label 1-925-925.264-r1-0-0
Degree $1$
Conductor $925$
Sign $-0.987 - 0.154i$
Analytic cond. $99.4050$
Root an. cond. $99.4050$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.927 − 0.374i)2-s + (−0.241 − 0.970i)3-s + (0.719 − 0.694i)4-s + (−0.587 − 0.809i)6-s + (0.939 − 0.342i)7-s + (0.406 − 0.913i)8-s + (−0.882 + 0.469i)9-s + (0.978 − 0.207i)11-s + (−0.848 − 0.529i)12-s + (−0.139 − 0.990i)13-s + (0.743 − 0.669i)14-s + (0.0348 − 0.999i)16-s + (−0.694 + 0.719i)17-s + (−0.642 + 0.766i)18-s + (0.0697 − 0.997i)19-s + ⋯
L(s)  = 1  + (0.927 − 0.374i)2-s + (−0.241 − 0.970i)3-s + (0.719 − 0.694i)4-s + (−0.587 − 0.809i)6-s + (0.939 − 0.342i)7-s + (0.406 − 0.913i)8-s + (−0.882 + 0.469i)9-s + (0.978 − 0.207i)11-s + (−0.848 − 0.529i)12-s + (−0.139 − 0.990i)13-s + (0.743 − 0.669i)14-s + (0.0348 − 0.999i)16-s + (−0.694 + 0.719i)17-s + (−0.642 + 0.766i)18-s + (0.0697 − 0.997i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.987 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.987 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.987 - 0.154i$
Analytic conductor: \(99.4050\)
Root analytic conductor: \(99.4050\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (264, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 925,\ (1:\ ),\ -0.987 - 0.154i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3229133696 - 4.154960239i\)
\(L(\frac12)\) \(\approx\) \(0.3229133696 - 4.154960239i\)
\(L(1)\) \(\approx\) \(1.394698726 - 1.410712783i\)
\(L(1)\) \(\approx\) \(1.394698726 - 1.410712783i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.927 - 0.374i)T \)
3 \( 1 + (-0.241 - 0.970i)T \)
7 \( 1 + (0.939 - 0.342i)T \)
11 \( 1 + (0.978 - 0.207i)T \)
13 \( 1 + (-0.139 - 0.990i)T \)
17 \( 1 + (-0.694 + 0.719i)T \)
19 \( 1 + (0.0697 - 0.997i)T \)
23 \( 1 + (0.743 - 0.669i)T \)
29 \( 1 + (0.994 + 0.104i)T \)
31 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-0.990 + 0.139i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.913 + 0.406i)T \)
53 \( 1 + (-0.559 - 0.829i)T \)
59 \( 1 + (0.999 + 0.0348i)T \)
61 \( 1 + (-0.469 + 0.882i)T \)
67 \( 1 + (0.559 - 0.829i)T \)
71 \( 1 + (-0.241 - 0.970i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.275 + 0.961i)T \)
83 \( 1 + (0.719 + 0.694i)T \)
89 \( 1 + (0.788 + 0.615i)T \)
97 \( 1 + (-0.406 - 0.913i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.82798011282349120474354421679, −21.57928282686260741192407904429, −20.68328868716986785872504475368, −20.147938536038874464453667228737, −18.98070901679268521647692684820, −17.638468528954945456443227458379, −17.15713124568049529746897834911, −16.32634032555158290295399238975, −15.603478729397976869926881293636, −14.74392264243472454636888593957, −14.34779587290653977327298119611, −13.51118204835964811003421857455, −12.09845784483411276963977372557, −11.64408712700857016751547656263, −11.10342777965197895729177358108, −9.83822119726435794302656828586, −8.93848322858063074415498723091, −8.10727953924728188466931154957, −6.9034264005145828860051566729, −6.14155774619284074065450304990, −5.103373359762239259923711119166, −4.527964316221778628473271891374, −3.80284508227892759903808257194, −2.662458367162756718707521226723, −1.515131610398226354161591705376, 0.64999557457313559562469164346, 1.40561254520246811030512921433, 2.40063543113879210938250873071, 3.39223363531886478920764275260, 4.65320398859350842656668557635, 5.24108139588001964439408912678, 6.471009083881017445506210700482, 6.850728065711479722676380728665, 8.01092000641260117569903869768, 8.83359575635725427909973509834, 10.39202553452279698249354506230, 11.001270579187209299734219126, 11.73730817813316413036128170812, 12.4574922211490283320921559625, 13.28343561695319027572175016305, 13.91795343353613431984221770188, 14.69855431925603847801831144412, 15.38871099700791157459339623010, 16.66169877988257504615180770183, 17.474454708993785487333475431196, 18.04942531909673469667702070675, 19.29372956456013508133150199097, 19.71231540883132846416533364656, 20.43750121723217149132276290067, 21.39476677713174532747357306800

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