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Properties

Label 1-65e2-4225.1138-r0-0-0
Degree $1$
Conductor $4225$
Sign $0.247 - 0.968i$
Analytic cond. $19.6208$
Root an. cond. $19.6208$
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  + (0.818 + 0.574i)2-s + (0.910 − 0.414i)3-s + (0.339 + 0.940i)4-s + (0.982 + 0.184i)6-s + (−0.987 − 0.160i)7-s + (−0.262 + 0.964i)8-s + (0.657 − 0.753i)9-s + (−0.753 + 0.657i)11-s + (0.698 + 0.715i)12-s + (−0.715 − 0.698i)14-s + (−0.769 + 0.638i)16-s + (0.709 − 0.704i)17-s + (0.970 − 0.239i)18-s + (−0.406 − 0.913i)19-s + (−0.964 + 0.262i)21-s + (−0.994 + 0.104i)22-s + ⋯
L(s)  = 1  + (0.818 + 0.574i)2-s + (0.910 − 0.414i)3-s + (0.339 + 0.940i)4-s + (0.982 + 0.184i)6-s + (−0.987 − 0.160i)7-s + (−0.262 + 0.964i)8-s + (0.657 − 0.753i)9-s + (−0.753 + 0.657i)11-s + (0.698 + 0.715i)12-s + (−0.715 − 0.698i)14-s + (−0.769 + 0.638i)16-s + (0.709 − 0.704i)17-s + (0.970 − 0.239i)18-s + (−0.406 − 0.913i)19-s + (−0.964 + 0.262i)21-s + (−0.994 + 0.104i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $0.247 - 0.968i$
Analytic conductor: \(19.6208\)
Root analytic conductor: \(19.6208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4225} (1138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4225,\ (0:\ ),\ 0.247 - 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.582585365 - 1.229464932i\)
\(L(\frac12)\) \(\approx\) \(1.582585365 - 1.229464932i\)
\(L(1)\) \(\approx\) \(1.663526364 + 0.1499048594i\)
\(L(1)\) \(\approx\) \(1.663526364 + 0.1499048594i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.818 + 0.574i)T \)
3 \( 1 + (0.910 - 0.414i)T \)
7 \( 1 + (-0.987 - 0.160i)T \)
11 \( 1 + (-0.753 + 0.657i)T \)
17 \( 1 + (0.709 - 0.704i)T \)
19 \( 1 + (-0.406 - 0.913i)T \)
23 \( 1 + (-0.743 - 0.669i)T \)
29 \( 1 + (0.293 - 0.955i)T \)
31 \( 1 + (-0.331 - 0.943i)T \)
37 \( 1 + (0.384 - 0.923i)T \)
41 \( 1 + (-0.493 + 0.870i)T \)
43 \( 1 + (0.0804 + 0.996i)T \)
47 \( 1 + (-0.926 + 0.377i)T \)
53 \( 1 + (-0.548 + 0.836i)T \)
59 \( 1 + (-0.968 + 0.247i)T \)
61 \( 1 + (0.541 - 0.840i)T \)
67 \( 1 + (0.339 - 0.940i)T \)
71 \( 1 + (-0.813 - 0.581i)T \)
73 \( 1 + (0.981 - 0.192i)T \)
79 \( 1 + (-0.926 + 0.377i)T \)
83 \( 1 + (-0.995 + 0.0965i)T \)
89 \( 1 + (0.743 + 0.669i)T \)
97 \( 1 + (-0.554 - 0.832i)T \)
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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

−18.90731892630484430583447185279, −18.25176128713834033440374371445, −16.88899978709454099087450252695, −16.00873193412261111193731832691, −15.89283590110156467310436519658, −14.95782741632897065001981179579, −14.380910451117830434642185094606, −13.74891960820904176503178973912, −13.091335702013528688757909539162, −12.598114898782411198306244901167, −11.87629616324626130203097812463, −10.81524539664051828841355784804, −10.14006799128056759488721145712, −9.97095591773366132233500792347, −8.88490120715327098349354588823, −8.29412257835936792810584693279, −7.33953352438381899715999307753, −6.48308086399012379290656737376, −5.65352522512757579711344363636, −5.11628792530997935500032435290, −3.98647413608134658231103359639, −3.463800132816534001150042003833, −3.02095798715922264222500140088, −2.10239505595002665414747107165, −1.321420334602721035847885020708, 0.31602223418109974069051720363, 1.87711073213059096494861415480, 2.765098517722614037040328755673, 3.01212551982237922628061077150, 4.132763346489775597952754663998, 4.58437548463083860344496750682, 5.70854562140001312571102416732, 6.45356509724992856968865108855, 6.99320572996252710300355676544, 7.83793744908023614289555365234, 8.10207054952008471562151230101, 9.367800284737636898447397944988, 9.63635152056169095023046239028, 10.70976754474643349108666413976, 11.718284853880191682764879112212, 12.53378453978311917757582151228, 12.879414949265331595003937605225, 13.51618424582541732523331774740, 14.08647795393746768507272065295, 14.86216051794522076142514840497, 15.4181843679952728578743586301, 16.01245382415586587601007663914, 16.633709589189913670566014384721, 17.55363218753253196614944750317, 18.25023811600112565193722202192

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