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Properties

Label 1-40e2-1600.669-r0-0-0
Degree $1$
Conductor $1600$
Sign $-0.191 - 0.981i$
Analytic cond. $7.43036$
Root an. cond. $7.43036$
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  + (−0.649 + 0.760i)3-s + (−0.707 + 0.707i)7-s + (−0.156 − 0.987i)9-s + (0.522 − 0.852i)11-s + (−0.852 + 0.522i)13-s + (0.951 − 0.309i)17-s + (0.996 + 0.0784i)19-s + (−0.0784 − 0.996i)21-s + (−0.987 − 0.156i)23-s + (0.852 + 0.522i)27-s + (−0.649 + 0.760i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)33-s + (−0.233 + 0.972i)37-s + (0.156 − 0.987i)39-s + ⋯
L(s)  = 1  + (−0.649 + 0.760i)3-s + (−0.707 + 0.707i)7-s + (−0.156 − 0.987i)9-s + (0.522 − 0.852i)11-s + (−0.852 + 0.522i)13-s + (0.951 − 0.309i)17-s + (0.996 + 0.0784i)19-s + (−0.0784 − 0.996i)21-s + (−0.987 − 0.156i)23-s + (0.852 + 0.522i)27-s + (−0.649 + 0.760i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)33-s + (−0.233 + 0.972i)37-s + (0.156 − 0.987i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.191 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.191 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.191 - 0.981i$
Analytic conductor: \(7.43036\)
Root analytic conductor: \(7.43036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (669, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1600,\ (0:\ ),\ -0.191 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1668326192 - 0.2024741339i\)
\(L(\frac12)\) \(\approx\) \(0.1668326192 - 0.2024741339i\)
\(L(1)\) \(\approx\) \(0.6528227984 + 0.1382240726i\)
\(L(1)\) \(\approx\) \(0.6528227984 + 0.1382240726i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.649 + 0.760i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (0.522 - 0.852i)T \)
13 \( 1 + (-0.852 + 0.522i)T \)
17 \( 1 + (0.951 - 0.309i)T \)
19 \( 1 + (0.996 + 0.0784i)T \)
23 \( 1 + (-0.987 - 0.156i)T \)
29 \( 1 + (-0.649 + 0.760i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.233 + 0.972i)T \)
41 \( 1 + (0.156 + 0.987i)T \)
43 \( 1 + (-0.923 + 0.382i)T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (-0.0784 - 0.996i)T \)
59 \( 1 + (-0.233 + 0.972i)T \)
61 \( 1 + (-0.972 + 0.233i)T \)
67 \( 1 + (0.0784 - 0.996i)T \)
71 \( 1 + (-0.453 - 0.891i)T \)
73 \( 1 + (0.987 + 0.156i)T \)
79 \( 1 + (-0.951 - 0.309i)T \)
83 \( 1 + (-0.996 - 0.0784i)T \)
89 \( 1 + (0.156 - 0.987i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−20.36158767015820637254255974337, −19.88594211014855371720431549680, −19.24102281043889979913559969791, −18.40883583367278631789285051131, −17.547776851919857876525296313175, −17.15160417757066504186684761156, −16.36059690993645256469751121185, −15.62593099477802443708867537216, −14.458820387898177121802577991886, −13.91470285177901664598571329120, −12.966901900831196992981957790276, −12.34407953355915824022633767105, −11.8707235865700890687142807504, −10.76965783429364376781410003900, −10.020117213020961866824152729942, −9.47078755184057195962999448284, −8.04275352008626064198740498882, −7.377983809576196537445769128607, −6.896534230352860609814220640639, −5.88066629751551473484045470185, −5.20904371531398656613983938716, −4.13570611932501758916241292888, −3.18675677444824586186650072182, −2.03029481615939287701268103798, −1.11209122233860381962045509588, 0.11586063698172944709235281980, 1.54203952799334816140916671719, 3.01249149808386421573608588515, 3.49203252777047626708760096482, 4.619091645512374938706782518173, 5.46190878705091481442163067323, 6.07344675716806383545789509295, 6.86070506348367725599684853979, 7.988374700030128633280068928117, 9.04403753225107154268461162280, 9.66208550108529951247914246775, 10.12284727985191904025315831879, 11.40483582782393200485255324493, 11.77785602803567220868298518333, 12.45011955576030762820147597183, 13.52215845057825712119574558671, 14.48092333233887886099774053159, 15.018890760544462400746096671441, 16.045116119214923232846113799456, 16.51099486673597440812649435828, 16.938386575104973839617775551347, 18.22159640077789414467107390790, 18.56095842355407683107814405222, 19.610860625756640021896650437832, 20.22874445590413922273938206239

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