↑

Properties

Label 1-903-903.125-r1-0-0
Degree $1$
Conductor $903$
Sign $0.993 - 0.117i$
Analytic cond. $97.0408$
Root an. cond. $97.0408$
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  + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s + (0.623 + 0.781i)8-s + (−0.623 + 0.781i)10-s + (0.900 + 0.433i)11-s + (−0.623 − 0.781i)13-s + (0.623 − 0.781i)16-s + (0.623 − 0.781i)17-s + (−0.900 + 0.433i)19-s + (0.900 + 0.433i)20-s + (0.222 − 0.974i)22-s + (0.900 + 0.433i)23-s + (−0.222 + 0.974i)25-s + (−0.623 + 0.781i)26-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s + (0.623 + 0.781i)8-s + (−0.623 + 0.781i)10-s + (0.900 + 0.433i)11-s + (−0.623 − 0.781i)13-s + (0.623 − 0.781i)16-s + (0.623 − 0.781i)17-s + (−0.900 + 0.433i)19-s + (0.900 + 0.433i)20-s + (0.222 − 0.974i)22-s + (0.900 + 0.433i)23-s + (−0.222 + 0.974i)25-s + (−0.623 + 0.781i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 903 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 903 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(903\)    =    \(3 \cdot 7 \cdot 43\)
Sign: $0.993 - 0.117i$
Analytic conductor: \(97.0408\)
Root analytic conductor: \(97.0408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{903} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 903,\ (1:\ ),\ 0.993 - 0.117i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8577034072 - 0.05075735191i\)
\(L(\frac12)\) \(\approx\) \(0.8577034072 - 0.05075735191i\)
\(L(1)\) \(\approx\) \(0.6510573878 - 0.3614463974i\)
\(L(1)\) \(\approx\) \(0.6510573878 - 0.3614463974i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (-0.623 - 0.781i)T \)
11 \( 1 + (0.900 + 0.433i)T \)
13 \( 1 + (-0.623 - 0.781i)T \)
17 \( 1 + (0.623 - 0.781i)T \)
19 \( 1 + (-0.900 + 0.433i)T \)
23 \( 1 + (0.900 + 0.433i)T \)
29 \( 1 + (-0.222 - 0.974i)T \)
31 \( 1 + (0.222 + 0.974i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.900 + 0.433i)T \)
53 \( 1 + (-0.623 + 0.781i)T \)
59 \( 1 + (0.623 - 0.781i)T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 + (-0.900 + 0.433i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.222 - 0.974i)T \)
97 \( 1 + (0.900 + 0.433i)T \)
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

−22.04064534861057310256320201343, −21.214144571692693070914863835887, −19.6784026227049314465982399011, −19.22780472025637639560353973698, −18.68447875146729389724360518158, −17.64642408557000591792647182423, −16.81272998539263044525733573090, −16.369670672681202389266109828676, −15.08658513414456475930254491747, −14.81123400068455490819594827307, −14.100529579169026960684458495513, −13.05427991448017475450976573119, −12.05114925523131090956783101018, −11.091023708133256580820685918989, −10.275572503984682738125586387347, −9.26028885630625047300494941183, −8.5001123808972402856665027127, −7.62305575538263416227910074658, −6.69352315928159244263370276435, −6.33992294804425901930885567888, −4.98765678375072263809145418253, −4.110059689202042562613705013104, −3.24542456236293431745065325427, −1.69055985870291306567448301407, −0.284504622937859327850046343462, 0.77874137186882461195519183890, 1.68664237813359183427058295100, 2.95897300599080185702163621752, 3.85197239183199144141042143338, 4.70897894887616531049208152923, 5.46651420465366641262279772294, 7.0702058219368256832168014406, 7.91374115677206087012021742965, 8.75397296340124833667986553576, 9.490089182074472316291345533007, 10.29841642817679759481204131969, 11.31324240047873167787171137464, 12.13479417093593451527370950399, 12.49963137198129377329696375974, 13.43513231581749900726133840426, 14.429184778896095558035577484126, 15.27205884820686227257551273600, 16.370102516364424042653959422206, 17.22182603611994015834928042550, 17.6085833269299721232154145681, 19.01567932854842260390257219799, 19.302122796780071064071787545868, 20.18998640002268537164459290208, 20.77109735279570682455652095049, 21.47078880898750071062391813640

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