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Properties

Label 1-633-633.143-r1-0-0
Degree $1$
Conductor $633$
Sign $-0.934 + 0.357i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.963 − 0.266i)2-s + (0.858 − 0.512i)4-s + (0.550 − 0.834i)5-s + (−0.963 − 0.266i)7-s + (0.691 − 0.722i)8-s + (0.309 − 0.951i)10-s + (−0.936 − 0.351i)11-s + (0.473 − 0.880i)13-s − 14-s + (0.473 − 0.880i)16-s + (−0.983 + 0.178i)17-s + (−0.809 + 0.587i)19-s + (0.0448 − 0.998i)20-s + (−0.995 − 0.0896i)22-s + (0.809 + 0.587i)23-s + ⋯
L(s)  = 1  + (0.963 − 0.266i)2-s + (0.858 − 0.512i)4-s + (0.550 − 0.834i)5-s + (−0.963 − 0.266i)7-s + (0.691 − 0.722i)8-s + (0.309 − 0.951i)10-s + (−0.936 − 0.351i)11-s + (0.473 − 0.880i)13-s − 14-s + (0.473 − 0.880i)16-s + (−0.983 + 0.178i)17-s + (−0.809 + 0.587i)19-s + (0.0448 − 0.998i)20-s + (−0.995 − 0.0896i)22-s + (0.809 + 0.587i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.934 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.934 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(633\)    =    \(3 \cdot 211\)
Sign: $-0.934 + 0.357i$
Analytic conductor: \(68.0252\)
Root analytic conductor: \(68.0252\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{633} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 633,\ (1:\ ),\ -0.934 + 0.357i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3048898738 - 1.651071471i\)
\(L(\frac12)\) \(\approx\) \(-0.3048898738 - 1.651071471i\)
\(L(1)\) \(\approx\) \(1.313230135 - 0.7728879616i\)
\(L(1)\) \(\approx\) \(1.313230135 - 0.7728879616i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.963 - 0.266i)T \)
5 \( 1 + (0.550 - 0.834i)T \)
7 \( 1 + (-0.963 - 0.266i)T \)
11 \( 1 + (-0.936 - 0.351i)T \)
13 \( 1 + (0.473 - 0.880i)T \)
17 \( 1 + (-0.983 + 0.178i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (-0.473 + 0.880i)T \)
31 \( 1 + (-0.900 - 0.433i)T \)
37 \( 1 + (0.936 - 0.351i)T \)
41 \( 1 + (-0.134 + 0.990i)T \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (0.393 + 0.919i)T \)
53 \( 1 + (-0.858 - 0.512i)T \)
59 \( 1 + (-0.134 + 0.990i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.623 + 0.781i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (-0.222 - 0.974i)T \)
79 \( 1 + (-0.691 - 0.722i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (0.0448 - 0.998i)T \)
97 \( 1 + (-0.995 - 0.0896i)T \)
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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

−23.16362327305698773767345818332, −22.3460668708226476972372333969, −21.736893618587054264430198598149, −21.03819422041222616272320609207, −20.080950180811627233816130599492, −19.04186218041426498238099093228, −18.32628287467118589566171677455, −17.23125704045852034368817158458, −16.385127595710188781076744789221, −15.42720697218469439622063297688, −15.000724817804241612719882841697, −13.82536954560605108199028236181, −13.240655383169853210802902125502, −12.59815274378956541457668512005, −11.29874676464394595454549581875, −10.74495668153643089475038195237, −9.64334118309380721563642725937, −8.58521000342441959840925936367, −7.157020460529230758910047752189, −6.66077799147509641762183806946, −5.89230601709906044158627238977, −4.80720804647607652657593586380, −3.728095396901185073203939273604, −2.658338600565063442336153789975, −2.10023397906794681297691900874, 0.2430869911174073982873679221, 1.513994017118145124171137283507, 2.691728906079702476675305373894, 3.61009353144222505556019053433, 4.6830623581461997831002825462, 5.65504294939052206399997132012, 6.20920783698806892635523754058, 7.395821173643626216063501520545, 8.57794386732176010590632015372, 9.65780817119526360627584967529, 10.51323613390412007887704741513, 11.23219819316116403272496926832, 12.73256003178854717102645180643, 12.96269693340528261879729549238, 13.46796186961973127554237759986, 14.69929787861631970923800809140, 15.64462332256751762880298110672, 16.263190789678506719619047323897, 17.05831668748634333452983949884, 18.25549984796905031516425575365, 19.22970963467748163617255388932, 20.18678471409178664377710244683, 20.57002951908696267049249686694, 21.58696748341204596940888080014, 22.12348028249104085448139357062

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