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Properties

Label 1-1601-1601.248-r0-0-0
Degree $1$
Conductor $1601$
Sign $0.997 + 0.0635i$
Analytic cond. $7.43501$
Root an. cond. $7.43501$
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  + (0.951 − 0.309i)2-s + (−0.233 + 0.972i)3-s + (0.809 − 0.587i)4-s + (0.587 + 0.809i)5-s + (0.0784 + 0.996i)6-s + (0.0784 − 0.996i)7-s + (0.587 − 0.809i)8-s + (−0.891 − 0.453i)9-s + (0.809 + 0.587i)10-s + (0.996 − 0.0784i)11-s + (0.382 + 0.923i)12-s + (−0.382 − 0.923i)13-s + (−0.233 − 0.972i)14-s + (−0.923 + 0.382i)15-s + (0.309 − 0.951i)16-s + (−0.649 + 0.760i)17-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (−0.233 + 0.972i)3-s + (0.809 − 0.587i)4-s + (0.587 + 0.809i)5-s + (0.0784 + 0.996i)6-s + (0.0784 − 0.996i)7-s + (0.587 − 0.809i)8-s + (−0.891 − 0.453i)9-s + (0.809 + 0.587i)10-s + (0.996 − 0.0784i)11-s + (0.382 + 0.923i)12-s + (−0.382 − 0.923i)13-s + (−0.233 − 0.972i)14-s + (−0.923 + 0.382i)15-s + (0.309 − 0.951i)16-s + (−0.649 + 0.760i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1601\)
Sign: $0.997 + 0.0635i$
Analytic conductor: \(7.43501\)
Root analytic conductor: \(7.43501\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1601} (248, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1601,\ (0:\ ),\ 0.997 + 0.0635i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.233003149 + 0.1028803086i\)
\(L(\frac12)\) \(\approx\) \(3.233003149 + 0.1028803086i\)
\(L(1)\) \(\approx\) \(2.001365064 + 0.08781894441i\)
\(L(1)\) \(\approx\) \(2.001365064 + 0.08781894441i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1601 \( 1 \)
good2 \( 1 + (0.951 - 0.309i)T \)
3 \( 1 + (-0.233 + 0.972i)T \)
5 \( 1 + (0.587 + 0.809i)T \)
7 \( 1 + (0.0784 - 0.996i)T \)
11 \( 1 + (0.996 - 0.0784i)T \)
13 \( 1 + (-0.382 - 0.923i)T \)
17 \( 1 + (-0.649 + 0.760i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.233 - 0.972i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (0.987 - 0.156i)T \)
41 \( 1 + (0.987 - 0.156i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (0.987 - 0.156i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.996 - 0.0784i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (-0.707 - 0.707i)T \)
71 \( 1 + (0.972 + 0.233i)T \)
73 \( 1 + (0.233 + 0.972i)T \)
79 \( 1 + T \)
83 \( 1 + (0.923 - 0.382i)T \)
89 \( 1 + (-0.987 + 0.156i)T \)
97 \( 1 + (-0.156 - 0.987i)T \)
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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

−20.51004015532985623316304168240, −19.75178901536313060940421704650, −19.16560134781068623451588170369, −17.89465306789884462410094033445, −17.54217752574776625070322021788, −16.68516362862081204335672973844, −16.03996054126010532748790833618, −15.11418873338925708707728624027, −14.16574941161412656534411323397, −13.694005122839035226291550541013, −12.99020123994340601083523100820, −12.156862935765466667503547465061, −11.768849157981917068099854367276, −11.13776614779676265197286126178, −9.33166996285857349268815675803, −9.04101642264934455880358056524, −7.92385864459856784109843867391, −7.06418086386191241190375485627, −6.31860105508870866722729921866, −5.69573481518694233412624083174, −4.93166398557898492212986019326, −4.13176805504359773473716382382, −2.57198151576627021023208903499, −2.195034558402357982130233474973, −1.14538481232127215076965422283, 1.02222476940342786661202218290, 2.25838726397811654337686502661, 3.256402874876681441959830712812, 3.8631539502397642213322687710, 4.58198374679910191452071028319, 5.59553119406750925291123719990, 6.27973716338836456385433702937, 6.934292960386335553058661401285, 8.07329148151910104676026152835, 9.441783991014190211669704814519, 10.07003346491128361828669037460, 10.838140989338688482442609028765, 10.98971211378511203840975119085, 12.21551323143214963218884311575, 12.92672143382689572555001113343, 14.10333200461763321708565588092, 14.38341267726858770670998271865, 14.915194199285782307325507187297, 15.921678167152743331802637910, 16.63543934408370706943691850483, 17.36745855296700965285439864807, 18.10298876276010090641915490967, 19.414558653970446064928771929567, 19.96191526635003870877543143770, 20.59104506857634973872826073822

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