↑

Properties

Label 1-633-633.254-r1-0-0
Degree $1$
Conductor $633$
Sign $-0.990 + 0.139i$
Analytic cond. $68.0252$
Root an. cond. $68.0252$
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  + (0.733 − 0.680i)2-s + (0.0747 − 0.997i)4-s + (0.222 − 0.974i)5-s + (0.955 − 0.294i)7-s + (−0.623 − 0.781i)8-s + (−0.5 − 0.866i)10-s + (0.222 + 0.974i)11-s + (0.623 − 0.781i)13-s + (0.5 − 0.866i)14-s + (−0.988 − 0.149i)16-s + (−0.365 + 0.930i)17-s + (−0.5 − 0.866i)19-s + (−0.955 − 0.294i)20-s + (0.826 + 0.563i)22-s − 23-s + ⋯
L(s)  = 1  + (0.733 − 0.680i)2-s + (0.0747 − 0.997i)4-s + (0.222 − 0.974i)5-s + (0.955 − 0.294i)7-s + (−0.623 − 0.781i)8-s + (−0.5 − 0.866i)10-s + (0.222 + 0.974i)11-s + (0.623 − 0.781i)13-s + (0.5 − 0.866i)14-s + (−0.988 − 0.149i)16-s + (−0.365 + 0.930i)17-s + (−0.5 − 0.866i)19-s + (−0.955 − 0.294i)20-s + (0.826 + 0.563i)22-s − 23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(633\)    =    \(3 \cdot 211\)
Sign: $-0.990 + 0.139i$
Analytic conductor: \(68.0252\)
Root analytic conductor: \(68.0252\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{633} (254, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 633,\ (1:\ ),\ -0.990 + 0.139i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2055969237 - 2.943815090i\)
\(L(\frac12)\) \(\approx\) \(-0.2055969237 - 2.943815090i\)
\(L(1)\) \(\approx\) \(1.172222167 - 1.190194089i\)
\(L(1)\) \(\approx\) \(1.172222167 - 1.190194089i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.733 - 0.680i)T \)
5 \( 1 + (0.222 - 0.974i)T \)
7 \( 1 + (0.955 - 0.294i)T \)
11 \( 1 + (0.222 + 0.974i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
17 \( 1 + (-0.365 + 0.930i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.365 - 0.930i)T \)
31 \( 1 + (0.365 - 0.930i)T \)
37 \( 1 + (0.955 - 0.294i)T \)
41 \( 1 + (0.988 - 0.149i)T \)
43 \( 1 + (-0.988 + 0.149i)T \)
47 \( 1 + (-0.0747 - 0.997i)T \)
53 \( 1 + (-0.0747 - 0.997i)T \)
59 \( 1 + (-0.365 + 0.930i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.222 + 0.974i)T \)
71 \( 1 - T \)
73 \( 1 + (0.0747 - 0.997i)T \)
79 \( 1 + (0.623 - 0.781i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
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

−23.27323731360615685123381751368, −22.304882688390295365753990338698, −21.60816934458650632342641289328, −21.1559421021155428109379118768, −20.075334273686204200328838041598, −18.62670589446272919473437427071, −18.28349039266698153748493669583, −17.30196678589651080661451291326, −16.33204202336015462072454001451, −15.65045370866585262291462079323, −14.48436668953461040684866116664, −14.23154772313027593998732909747, −13.49234059683934519317888362664, −12.17965744216214721025053016804, −11.38015741123088802061847273448, −10.81408174345368363429056612658, −9.277889758770258901095537175676, −8.35291916708983035235592640221, −7.55418333320339845422527754501, −6.44839256662989787366531312702, −5.94433897681993124601019991955, −4.79192927955774362423006084620, −3.79191182869781758621623678798, −2.83517375180192712118204594870, −1.7061616108639838947236314175, 0.50243241815364635586554614554, 1.58983903269969045141806874739, 2.323215676532535416377243308148, 4.06757487979431364085644337409, 4.40982603520931428148602401807, 5.473324585529429449581079468837, 6.28862698650395334671476896155, 7.72789213393881835671078761922, 8.647357314535942444759817076968, 9.69073021135329421932374794577, 10.51746049238998284118861155594, 11.4459448509991319572855875972, 12.2154680594500421435646447074, 13.174550759636546872880414191, 13.53981528618494504837898214517, 14.896507019117065883376772754342, 15.222503316972476192346574061, 16.46684125289459160003626556265, 17.6116262122598813103687882921, 17.972379536060916642079499456845, 19.43429849566310808410737972678, 20.09887065406377233511906687725, 20.67923474747940220572339179096, 21.33526803141724519627115834794, 22.19388934385577336877954308665

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