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Properties

Label 1-40e2-1600.227-r0-0-0
Degree $1$
Conductor $1600$
Sign $-0.687 - 0.726i$
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  + (0.852 − 0.522i)3-s + (−0.707 − 0.707i)7-s + (0.453 − 0.891i)9-s + (−0.0784 − 0.996i)11-s + (−0.0784 + 0.996i)13-s + (0.809 − 0.587i)17-s + (0.233 + 0.972i)19-s + (−0.972 − 0.233i)21-s + (−0.453 − 0.891i)23-s + (−0.0784 − 0.996i)27-s + (−0.522 − 0.852i)29-s + (−0.809 + 0.587i)31-s + (−0.587 − 0.809i)33-s + (−0.649 − 0.760i)37-s + (0.453 + 0.891i)39-s + ⋯
L(s)  = 1  + (0.852 − 0.522i)3-s + (−0.707 − 0.707i)7-s + (0.453 − 0.891i)9-s + (−0.0784 − 0.996i)11-s + (−0.0784 + 0.996i)13-s + (0.809 − 0.587i)17-s + (0.233 + 0.972i)19-s + (−0.972 − 0.233i)21-s + (−0.453 − 0.891i)23-s + (−0.0784 − 0.996i)27-s + (−0.522 − 0.852i)29-s + (−0.809 + 0.587i)31-s + (−0.587 − 0.809i)33-s + (−0.649 − 0.760i)37-s + (0.453 + 0.891i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.687 - 0.726i$
Analytic conductor: \(7.43036\)
Root analytic conductor: \(7.43036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1600,\ (0:\ ),\ -0.687 - 0.726i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6214136782 - 1.443773370i\)
\(L(\frac12)\) \(\approx\) \(0.6214136782 - 1.443773370i\)
\(L(1)\) \(\approx\) \(1.106112839 - 0.5231239406i\)
\(L(1)\) \(\approx\) \(1.106112839 - 0.5231239406i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.852 - 0.522i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (-0.0784 - 0.996i)T \)
13 \( 1 + (-0.0784 + 0.996i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (0.233 + 0.972i)T \)
23 \( 1 + (-0.453 - 0.891i)T \)
29 \( 1 + (-0.522 - 0.852i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.649 - 0.760i)T \)
41 \( 1 + (0.453 - 0.891i)T \)
43 \( 1 + (0.382 + 0.923i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (0.233 - 0.972i)T \)
59 \( 1 + (0.760 - 0.649i)T \)
61 \( 1 + (0.649 - 0.760i)T \)
67 \( 1 + (0.233 + 0.972i)T \)
71 \( 1 + (0.987 - 0.156i)T \)
73 \( 1 + (-0.453 - 0.891i)T \)
79 \( 1 + (-0.587 + 0.809i)T \)
83 \( 1 + (-0.972 + 0.233i)T \)
89 \( 1 + (-0.453 - 0.891i)T \)
97 \( 1 + (-0.587 + 0.809i)T \)
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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

−20.634953289688045127904817267567, −19.96899124214719337100470072631, −19.486001533966952818600817281492, −18.599943684413675783830874227082, −17.88056643614528589463965886511, −16.9404703292572925759160871501, −16.0460227035495732522115602348, −15.33059869232238820576327499487, −15.02346003165892593097178876701, −14.11315395951354609301985066778, −13.025661064998538604838404153199, −12.76991470899068057980018565181, −11.71676768342141796702058328411, −10.6221743453532537984376485993, −9.88417034063383115745206385947, −9.39534576449223997263042480263, −8.554741977442469571122259812721, −7.688835664346504486877242589183, −7.02482140283473892559598670557, −5.7193048969257245358866042209, −5.139135972229726610431897724192, −4.01824069651658473219510045454, −3.20949486263337123202346728673, −2.53382405536697530930577641483, −1.5144991705718477732115800270, 0.49375299858805204783952746831, 1.60586366527888358048931225458, 2.60719165939902685704085199311, 3.60458068046587004768860388702, 3.974511587339877581687767943215, 5.447926857224492693865959939681, 6.39373252241583215046506886638, 7.05887965237657130798656960217, 7.85211667726373091202046668741, 8.60371907879754841635031197582, 9.49846846390549390132479521417, 10.04482479496312158182829826361, 11.109694343595441540767826258063, 12.04127299550797363338695366302, 12.761011433917895543278937505410, 13.52366582239362168108232136596, 14.31867252095610902454712804049, 14.44462322303989409961003345721, 16.05952937101310190494620449694, 16.20988665472312462793064366274, 17.15740402784899845859818299690, 18.25562861921574739327043844024, 18.910932007382202958704066981051, 19.295950516538908003766857461917, 20.14854104639995316920688678934

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