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Properties

Label 1-40e2-1600.379-r1-0-0
Degree $1$
Conductor $1600$
Sign $-0.558 - 0.829i$
Analytic cond. $171.943$
Root an. cond. $171.943$
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  + (−0.760 + 0.649i)3-s + (−0.707 − 0.707i)7-s + (0.156 − 0.987i)9-s + (−0.852 + 0.522i)11-s + (0.522 − 0.852i)13-s + (0.951 + 0.309i)17-s + (−0.0784 − 0.996i)19-s + (0.996 + 0.0784i)21-s + (−0.987 + 0.156i)23-s + (0.522 + 0.852i)27-s + (0.760 − 0.649i)29-s + (0.309 − 0.951i)31-s + (0.309 − 0.951i)33-s + (−0.972 + 0.233i)37-s + (0.156 + 0.987i)39-s + ⋯
L(s)  = 1  + (−0.760 + 0.649i)3-s + (−0.707 − 0.707i)7-s + (0.156 − 0.987i)9-s + (−0.852 + 0.522i)11-s + (0.522 − 0.852i)13-s + (0.951 + 0.309i)17-s + (−0.0784 − 0.996i)19-s + (0.996 + 0.0784i)21-s + (−0.987 + 0.156i)23-s + (0.522 + 0.852i)27-s + (0.760 − 0.649i)29-s + (0.309 − 0.951i)31-s + (0.309 − 0.951i)33-s + (−0.972 + 0.233i)37-s + (0.156 + 0.987i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.558 - 0.829i$
Analytic conductor: \(171.943\)
Root analytic conductor: \(171.943\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1600,\ (1:\ ),\ -0.558 - 0.829i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3577581685 - 0.6724912746i\)
\(L(\frac12)\) \(\approx\) \(0.3577581685 - 0.6724912746i\)
\(L(1)\) \(\approx\) \(0.7259741853 - 0.03641549507i\)
\(L(1)\) \(\approx\) \(0.7259741853 - 0.03641549507i\)

Euler product

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

−20.71189856582647854672815131065, −19.34353639132544580632014610538, −19.06253226437932437381599774707, −18.29943573025819904251956281564, −17.82845363115210410005167962029, −16.54840622047162564769115704269, −16.26063315571000571259140518564, −15.64963958715150078123850382019, −14.27054373773070370677674919782, −13.77986634856040311220524990041, −12.82338901013194379809579132525, −12.16922853365406253593929231554, −11.739659392561276557056989485889, −10.55461452744939272378566040813, −10.14697885168377779408416059193, −8.90388421757136195295278688555, −8.21522106812373607072665995113, −7.288917523422164538620523592387, −6.441582721720845588583535297560, −5.75713014603906381237559611332, −5.20077963872265854231947917782, −3.92139212038365378834361232777, −2.89920626164850699639757320731, −1.96522337959854388649670971415, −0.932375153735381791283256530731, 0.22397474437743411608224626321, 0.9299217494385555927984257647, 2.51764760518457605961212907776, 3.51960748015417080904732491248, 4.198236306026375752508477471146, 5.19005383219326946600773117136, 5.87220008968058072399630176968, 6.71241656783011894619882331734, 7.58978873037260120173416024866, 8.48452890818017843688551342867, 9.71422115319220393191853344233, 10.14562766180058293283518277102, 10.68174706621363963701591786757, 11.64882100161985798859036770655, 12.465643651041509868997637353337, 13.16149470448594106070769474252, 13.92281448580549002115308737365, 15.12879903638206890335592984625, 15.63302963675212348312418943694, 16.20993101841376624027274872775, 17.13693072149445949809703164871, 17.59125381961268283454155638408, 18.42072476551955665375196002101, 19.29227271477394083337286040769, 20.33795062725884368706378571814

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