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Properties

Label 1-80e2-6400.637-r1-0-0
Degree $1$
Conductor $6400$
Sign $0.999 + 0.00687i$
Analytic cond. $687.775$
Root an. cond. $687.775$
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  + (0.252 − 0.967i)3-s + (−0.195 − 0.980i)7-s + (−0.872 − 0.488i)9-s + (0.400 + 0.916i)11-s + (0.0196 − 0.999i)13-s + (0.522 + 0.852i)17-s + (0.436 + 0.899i)19-s + (−0.998 − 0.0588i)21-s + (−0.785 − 0.619i)23-s + (−0.693 + 0.720i)27-s + (−0.505 − 0.862i)29-s + (0.987 − 0.156i)31-s + (0.987 − 0.156i)33-s + (0.175 + 0.984i)37-s + (−0.962 − 0.271i)39-s + ⋯
L(s)  = 1  + (0.252 − 0.967i)3-s + (−0.195 − 0.980i)7-s + (−0.872 − 0.488i)9-s + (0.400 + 0.916i)11-s + (0.0196 − 0.999i)13-s + (0.522 + 0.852i)17-s + (0.436 + 0.899i)19-s + (−0.998 − 0.0588i)21-s + (−0.785 − 0.619i)23-s + (−0.693 + 0.720i)27-s + (−0.505 − 0.862i)29-s + (0.987 − 0.156i)31-s + (0.987 − 0.156i)33-s + (0.175 + 0.984i)37-s + (−0.962 − 0.271i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6400\)    =    \(2^{8} \cdot 5^{2}\)
Sign: $0.999 + 0.00687i$
Analytic conductor: \(687.775\)
Root analytic conductor: \(687.775\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6400} (637, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6400,\ (1:\ ),\ 0.999 + 0.00687i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.549492481 + 0.005324258355i\)
\(L(\frac12)\) \(\approx\) \(1.549492481 + 0.005324258355i\)
\(L(1)\) \(\approx\) \(0.9726760190 - 0.3892729689i\)
\(L(1)\) \(\approx\) \(0.9726760190 - 0.3892729689i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.252 - 0.967i)T \)
7 \( 1 + (-0.195 - 0.980i)T \)
11 \( 1 + (0.400 + 0.916i)T \)
13 \( 1 + (0.0196 - 0.999i)T \)
17 \( 1 + (0.522 + 0.852i)T \)
19 \( 1 + (0.436 + 0.899i)T \)
23 \( 1 + (-0.785 - 0.619i)T \)
29 \( 1 + (-0.505 - 0.862i)T \)
31 \( 1 + (0.987 - 0.156i)T \)
37 \( 1 + (0.175 + 0.984i)T \)
41 \( 1 + (-0.271 - 0.962i)T \)
43 \( 1 + (-0.773 - 0.634i)T \)
47 \( 1 + (-0.233 - 0.972i)T \)
53 \( 1 + (-0.747 + 0.664i)T \)
59 \( 1 + (0.820 + 0.571i)T \)
61 \( 1 + (0.842 + 0.539i)T \)
67 \( 1 + (0.0588 + 0.998i)T \)
71 \( 1 + (-0.734 - 0.678i)T \)
73 \( 1 + (-0.993 + 0.117i)T \)
79 \( 1 + (-0.522 + 0.852i)T \)
83 \( 1 + (0.436 + 0.899i)T \)
89 \( 1 + (0.962 + 0.271i)T \)
97 \( 1 + (-0.987 - 0.156i)T \)
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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

−17.401238287978104440614627867666, −16.44625436527266486846844438038, −16.07031135660691378541552797464, −15.75957770761372092630831473790, −14.67797095698379297191417462397, −14.39275754225338938087235252319, −13.680063532961175128946321956421, −12.97581491775059972027628051447, −11.79750833335460568342366501810, −11.606642605075219830539347497136, −11.01182362789896926032283430460, −9.91321161311068942163726867106, −9.473166406302659653736709823277, −8.9644131141890766576639216877, −8.35872187105230815247642538642, −7.543092581313340796938192738840, −6.50922265789087769772042135104, −5.94854079829098966975037835431, −5.11986284025889164094812598367, −4.67367256103035342400364353362, −3.60815863556100208537795191705, −3.14743759300871962439995078002, −2.4206623916816529730895621819, −1.46507070213030959851468005435, −0.25803073842960173150978927356, 0.65472035618721407082987616844, 1.368223389394500462465491920795, 2.05675581632564588559826071693, 3.00489002892115114631292603410, 3.74456726118268192724030826081, 4.34446457673682125113042396987, 5.51701119511031907967117314247, 6.06564206555212710126728500574, 6.88108202950730440849577284820, 7.36374934239697417851875214315, 8.18361669047997769456249144678, 8.3920149323238120731933997253, 9.80612286167613697832123078131, 10.00462525031893574178361105375, 10.78091176023655884174243899878, 11.91351075201332407587221764963, 12.12845593974544225894915823549, 12.957202315730467550720947512863, 13.46612405246673484615031158244, 14.0800984071300059968966171145, 14.817297158618779673741826869298, 15.23085679370783047010070449136, 16.31144023524370164212250356844, 17.003822045993312890302474387162, 17.44027167678886877482212045460

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