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Properties

Label 1-629-629.174-r0-0-0
Degree $1$
Conductor $629$
Sign $0.00779 - 0.999i$
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  + (0.5 − 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s + i·6-s + (0.866 − 0.5i)7-s − 8-s + (0.5 − 0.866i)9-s + i·10-s + i·11-s + (0.866 + 0.5i)12-s + (−0.5 − 0.866i)13-s − i·14-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s + i·6-s + (0.866 − 0.5i)7-s − 8-s + (0.5 − 0.866i)9-s + i·10-s + i·11-s + (0.866 + 0.5i)12-s + (−0.5 − 0.866i)13-s − i·14-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 629 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00779 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 629 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00779 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(629\)    =    \(17 \cdot 37\)
Sign: $0.00779 - 0.999i$
Analytic conductor: \(2.92106\)
Root analytic conductor: \(2.92106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{629} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 629,\ (0:\ ),\ 0.00779 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7064799125 - 0.7120064256i\)
\(L(\frac12)\) \(\approx\) \(0.7064799125 - 0.7120064256i\)
\(L(1)\) \(\approx\) \(0.8217134608 - 0.3569735387i\)
\(L(1)\) \(\approx\) \(0.8217134608 - 0.3569735387i\)

Euler product

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

−23.594342921105549612856601210091, −22.416959555002247616816330140110, −21.79141622456737328098712016357, −21.0778446865725962836563755671, −19.7641699642258909362615886963, −18.74467900589735600263300133549, −18.11212634590288520618885968834, −17.1638281967049955092873971595, −16.39993052150647204044248261265, −15.94323733275357195890411088105, −14.84805955195032622134870015703, −14.01553846470975388571961845754, −13.05716354492735793044570936377, −12.12273946382357393214797301327, −11.701766992697385805224118371617, −10.836975172788062376498310424089, −8.987444435210674652787038442929, −8.415617834844879128204563453698, −7.43291484310363219735976569284, −6.74842962296377247028122406128, −5.58142739286482354448494025346, −4.92467303717738759776076085685, −4.19000196980379710337210306916, −2.70437221138482594789518243837, −1.01181263529798645072385210246, 0.618644256391683156486560385898, 1.99884214278705974045650355327, 3.45291281137216038544575592692, 4.17565576839644200263427479990, 4.95125580004245670481208085025, 5.82999441205842613208077616564, 7.154916947973842780645462682104, 7.98865984455209318112039499243, 9.61233883077414321754581352018, 10.23518770741081625605127732757, 10.9933955206298015571604415073, 11.77072051255448328885795015915, 12.229618268772450238170216681, 13.35343027091698624516409891528, 14.63034275550606037995469984924, 15.041874597725438262010998323598, 15.84834747563238892075067839745, 17.226713413294521980186198282462, 17.82299899115061817829051748401, 18.62656360444992110138956179433, 19.69160380708694080839736388594, 20.49101026748078895652206118314, 20.99845501967281924052096332570, 22.18215609631438203492467402452, 22.63317637920690573803886871380

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