↑

Properties

Label 1-203-203.188-r0-0-0
Degree $1$
Conductor $203$
Sign $-0.757 - 0.653i$
Analytic cond. $0.942728$
Root an. cond. $0.942728$
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  + (−0.974 + 0.222i)2-s + (0.433 − 0.900i)3-s + (0.900 − 0.433i)4-s + (−0.222 − 0.974i)5-s + (−0.222 + 0.974i)6-s + (−0.781 + 0.623i)8-s + (−0.623 − 0.781i)9-s + (0.433 + 0.900i)10-s + (−0.781 − 0.623i)11-s − i·12-s + (0.623 − 0.781i)13-s + (−0.974 − 0.222i)15-s + (0.623 − 0.781i)16-s + i·17-s + (0.781 + 0.623i)18-s + (−0.433 − 0.900i)19-s + ⋯
L(s)  = 1  + (−0.974 + 0.222i)2-s + (0.433 − 0.900i)3-s + (0.900 − 0.433i)4-s + (−0.222 − 0.974i)5-s + (−0.222 + 0.974i)6-s + (−0.781 + 0.623i)8-s + (−0.623 − 0.781i)9-s + (0.433 + 0.900i)10-s + (−0.781 − 0.623i)11-s − i·12-s + (0.623 − 0.781i)13-s + (−0.974 − 0.222i)15-s + (0.623 − 0.781i)16-s + i·17-s + (0.781 + 0.623i)18-s + (−0.433 − 0.900i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $-0.757 - 0.653i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ -0.757 - 0.653i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2300201813 - 0.6189502727i\)
\(L(\frac12)\) \(\approx\) \(0.2300201813 - 0.6189502727i\)
\(L(1)\) \(\approx\) \(0.5899424516 - 0.3705136756i\)
\(L(1)\) \(\approx\) \(0.5899424516 - 0.3705136756i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.974 + 0.222i)T \)
3 \( 1 + (0.433 - 0.900i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
11 \( 1 + (-0.781 - 0.623i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
17 \( 1 + iT \)
19 \( 1 + (-0.433 - 0.900i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
31 \( 1 + (0.974 - 0.222i)T \)
37 \( 1 + (-0.781 + 0.623i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.974 + 0.222i)T \)
47 \( 1 + (-0.781 - 0.623i)T \)
53 \( 1 + (-0.222 - 0.974i)T \)
59 \( 1 - T \)
61 \( 1 + (0.433 - 0.900i)T \)
67 \( 1 + (-0.623 - 0.781i)T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (0.974 + 0.222i)T \)
79 \( 1 + (0.781 - 0.623i)T \)
83 \( 1 + (0.900 - 0.433i)T \)
89 \( 1 + (0.974 - 0.222i)T \)
97 \( 1 + (0.433 + 0.900i)T \)
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

−27.02241578993453508288248898135, −26.37377509485984357499849582916, −25.78019131584264618174499713333, −24.85408388672733865738028578031, −23.25962743867423309264999942402, −22.35025964658046879459958597880, −21.101209105206159053585535295909, −20.70459533747583201608194006338, −19.48052310251986433024117897860, −18.678193217607019187077002841436, −17.843761934096440328714216178001, −16.47627146525377352478243630710, −15.80628341573240278477783193076, −14.89110060866987396886297033950, −13.8751592128578336863573326210, −12.168564861743547282919545093130, −11.03514924734668998794207796926, −10.37178994847341887096751011351, −9.50616456903316667988059574218, −8.35755032052479556176963481789, −7.4156327764293235726261038475, −6.19481804975520842530716396948, −4.37172506926087878485929830740, −3.1145109503504751123757245148, −2.18627347474340252790726976156, 0.634359210238486977699843982625, 1.87984080959754260052892872114, 3.317438342502427099415394016469, 5.40644979199435117284150656449, 6.40357006600292923038687780118, 7.83161224844127728384654154863, 8.3010319285803290948532219783, 9.181911183303638301134065778792, 10.59852619824952089835536148662, 11.71919324859571623158687184944, 12.82514775770057702693615862970, 13.64242379061292636436074131413, 15.21239076810362785946325523003, 15.87614917298984797617705333424, 17.20503241346164617679812535565, 17.76414883021166686655627456500, 18.976428085032129663287888214330, 19.56258345576308881545942624434, 20.49570519297108165234424565604, 21.265146679084030209288217490664, 23.28799521943506968808337567323, 24.00725019911197548590277719783, 24.575077410687913633353229415544, 25.68351897950483962417825408158, 26.179628902524912862140492495491

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