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Properties

Label 1-637-637.543-r0-0-0
Degree $1$
Conductor $637$
Sign $-0.825 + 0.565i$
Analytic cond. $2.95821$
Root an. cond. $2.95821$
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  + (0.988 + 0.149i)2-s + (−0.900 − 0.433i)3-s + (0.955 + 0.294i)4-s + (−0.826 + 0.563i)5-s + (−0.826 − 0.563i)6-s + (0.900 + 0.433i)8-s + (0.623 + 0.781i)9-s + (−0.900 + 0.433i)10-s + (−0.623 + 0.781i)11-s + (−0.733 − 0.680i)12-s + (0.988 − 0.149i)15-s + (0.826 + 0.563i)16-s + (−0.733 − 0.680i)17-s + (0.5 + 0.866i)18-s − 19-s + (−0.955 + 0.294i)20-s + ⋯
L(s)  = 1  + (0.988 + 0.149i)2-s + (−0.900 − 0.433i)3-s + (0.955 + 0.294i)4-s + (−0.826 + 0.563i)5-s + (−0.826 − 0.563i)6-s + (0.900 + 0.433i)8-s + (0.623 + 0.781i)9-s + (−0.900 + 0.433i)10-s + (−0.623 + 0.781i)11-s + (−0.733 − 0.680i)12-s + (0.988 − 0.149i)15-s + (0.826 + 0.563i)16-s + (−0.733 − 0.680i)17-s + (0.5 + 0.866i)18-s − 19-s + (−0.955 + 0.294i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.825 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.825 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.825 + 0.565i$
Analytic conductor: \(2.95821\)
Root analytic conductor: \(2.95821\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 637,\ (0:\ ),\ -0.825 + 0.565i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2275660056 + 0.7349605302i\)
\(L(\frac12)\) \(\approx\) \(0.2275660056 + 0.7349605302i\)
\(L(1)\) \(\approx\) \(0.9688283406 + 0.2728423746i\)
\(L(1)\) \(\approx\) \(0.9688283406 + 0.2728423746i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.988 + 0.149i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (-0.826 + 0.563i)T \)
11 \( 1 + (-0.623 + 0.781i)T \)
17 \( 1 + (-0.733 - 0.680i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.733 + 0.680i)T \)
29 \( 1 + (-0.733 - 0.680i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.955 + 0.294i)T \)
41 \( 1 + (-0.826 + 0.563i)T \)
43 \( 1 + (0.826 + 0.563i)T \)
47 \( 1 + (-0.365 - 0.930i)T \)
53 \( 1 + (-0.733 + 0.680i)T \)
59 \( 1 + (-0.0747 + 0.997i)T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 - T \)
71 \( 1 + (0.733 - 0.680i)T \)
73 \( 1 + (-0.365 + 0.930i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.623 - 0.781i)T \)
89 \( 1 + (-0.365 + 0.930i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−22.50665246457147882044587016438, −21.972299454557030871388576057067, −20.98556450826862133735868469951, −20.52073580823971059781655080347, −19.398512511666385251664997774987, −18.68652589650332056259342901848, −17.27323497376321406155742507701, −16.58488978466764259870896391324, −15.75294173879116014371953257021, −15.36826564976571457533641134262, −14.27672817595170463522183106294, −12.9735377245275131926390074810, −12.60941775550173785355825683532, −11.6108320943731862544628244003, −10.949001297468577455423571578350, −10.31501915099924451698341770549, −8.85560750410744654016936726811, −7.811360401050783053060184677719, −6.63111855593289980874080415125, −5.85081285031021539408777682981, −4.91927367196553917224758962966, −4.18536950987817294961210149374, −3.41025111407173172230247869530, −1.86787439410862805990424341983, −0.2843161493623785490447088148, 1.8065566251827522531756477907, 2.813686294106965157456532883318, 4.153598063803929201501044216977, 4.78650381195635587676922415623, 5.8631598787098170611438189200, 6.81209140718676119361650887201, 7.34915437657258929353667963629, 8.23655144743448351484880299616, 10.1237414951507175553792459734, 10.86875469587771999060350406710, 11.65969526206584974111738763046, 12.25715259283989773626400484150, 13.120850965996272420179836663732, 13.921875995637093741582745256, 15.15336716761796288347887250113, 15.588236918876155310346483693977, 16.42671760355418725671571038107, 17.44114753404890728295834063213, 18.22466518452910369393906460523, 19.23943134250151880393302898457, 19.972705719112424796361324313990, 21.02161211261313747456239607770, 21.91408168521696911221754165019, 22.71531223374225543172984245775, 23.122608776021405324592239479724

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