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Properties

Label 1-629-629.430-r0-0-0
Degree $1$
Conductor $629$
Sign $0.555 - 0.831i$
Analytic cond. $2.92106$
Root an. cond. $2.92106$
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  + (0.258 − 0.965i)2-s + (0.608 + 0.793i)3-s + (−0.866 − 0.5i)4-s + (0.608 + 0.793i)5-s + (0.923 − 0.382i)6-s + (0.130 − 0.991i)7-s + (−0.707 + 0.707i)8-s + (−0.258 + 0.965i)9-s + (0.923 − 0.382i)10-s + (−0.382 − 0.923i)11-s + (−0.130 − 0.991i)12-s + (0.5 − 0.866i)13-s + (−0.923 − 0.382i)14-s + (−0.258 + 0.965i)15-s + (0.5 + 0.866i)16-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (0.608 + 0.793i)3-s + (−0.866 − 0.5i)4-s + (0.608 + 0.793i)5-s + (0.923 − 0.382i)6-s + (0.130 − 0.991i)7-s + (−0.707 + 0.707i)8-s + (−0.258 + 0.965i)9-s + (0.923 − 0.382i)10-s + (−0.382 − 0.923i)11-s + (−0.130 − 0.991i)12-s + (0.5 − 0.866i)13-s + (−0.923 − 0.382i)14-s + (−0.258 + 0.965i)15-s + (0.5 + 0.866i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 629 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 629 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(629\)    =    \(17 \cdot 37\)
Sign: $0.555 - 0.831i$
Analytic conductor: \(2.92106\)
Root analytic conductor: \(2.92106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{629} (430, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 629,\ (0:\ ),\ 0.555 - 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.774018666 - 0.9485879743i\)
\(L(\frac12)\) \(\approx\) \(1.774018666 - 0.9485879743i\)
\(L(1)\) \(\approx\) \(1.414724563 - 0.4706000758i\)
\(L(1)\) \(\approx\) \(1.414724563 - 0.4706000758i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (0.608 + 0.793i)T \)
5 \( 1 + (0.608 + 0.793i)T \)
7 \( 1 + (0.130 - 0.991i)T \)
11 \( 1 + (-0.382 - 0.923i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.965 - 0.258i)T \)
23 \( 1 + (0.923 - 0.382i)T \)
29 \( 1 + (0.382 - 0.923i)T \)
31 \( 1 + (-0.923 - 0.382i)T \)
41 \( 1 + (-0.130 + 0.991i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.965 + 0.258i)T \)
59 \( 1 + (0.965 + 0.258i)T \)
61 \( 1 + (0.991 + 0.130i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.608 - 0.793i)T \)
73 \( 1 + (0.923 + 0.382i)T \)
79 \( 1 + (0.130 - 0.991i)T \)
83 \( 1 + (-0.965 - 0.258i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.382 + 0.923i)T \)
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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

−23.43373750093053224883369029002, −22.41740826705464543896263445082, −21.32914353213273304591489151028, −20.86598595972104113465962545562, −19.76233254864852616408490015626, −18.576598343500625441923959153813, −18.122906117096068549663937290024, −17.40294851748205952089666563426, −16.334688000753165871422230072309, −15.56326562040498988206029727045, −14.625454278697445697792892102782, −13.96754010828533090874068579009, −13.05086753341870769286741083561, −12.5323542872855277631174087814, −11.72751550610615928448513902876, −9.77561007790065209814899052066, −9.003756747940993542180093947657, −8.58292307368377627198531128129, −7.45617452706486760515523357179, −6.69737924749082820889154989261, −5.61786844659159989048050673479, −5.00496272519653770630313761546, −3.664864381448212762389047844498, −2.38246028313040204759599232043, −1.32784479897312355196523786835, 1.008402466918675634015541039910, 2.514072758810834030099090466330, 3.18431737456623314607458211059, 3.93113574092233242691074688332, 5.11565151496134931773647377436, 5.93101530497129816572240753693, 7.458891757794327183263357089222, 8.468431900343796113568067409755, 9.47830410042897363385394151455, 10.26012036614455721990941579883, 10.81902589699403142381415990783, 11.39529155052070488659447949692, 13.206280554490986845319398006550, 13.48503791945601303835166985756, 14.30593667812399267934642901587, 15.00355344729382886394654109675, 16.07805763860887142678829242055, 17.1864422699279172215219181837, 18.05582510033296647479196910106, 18.93201927855987220789548605770, 19.69602186736416074055588074497, 20.67050382925188252081899181006, 20.95197091679447229508244091927, 21.94983179852547684047356780726, 22.527443448111390109873249947550

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