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Properties

Label 1-4025-4025.223-r0-0-0
Degree $1$
Conductor $4025$
Sign $-0.825 + 0.564i$
Analytic cond. $18.6920$
Root an. cond. $18.6920$
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  + (−0.170 + 0.985i)2-s + (0.491 + 0.870i)3-s + (−0.941 − 0.336i)4-s + (−0.941 + 0.336i)6-s + (0.491 − 0.870i)8-s + (−0.516 + 0.856i)9-s + (0.897 + 0.441i)11-s + (−0.170 − 0.985i)12-s + (0.967 + 0.254i)13-s + (0.774 + 0.633i)16-s + (0.336 + 0.941i)17-s + (−0.755 − 0.654i)18-s + (0.610 − 0.791i)19-s + (−0.587 + 0.809i)22-s + 24-s + ⋯
L(s)  = 1  + (−0.170 + 0.985i)2-s + (0.491 + 0.870i)3-s + (−0.941 − 0.336i)4-s + (−0.941 + 0.336i)6-s + (0.491 − 0.870i)8-s + (−0.516 + 0.856i)9-s + (0.897 + 0.441i)11-s + (−0.170 − 0.985i)12-s + (0.967 + 0.254i)13-s + (0.774 + 0.633i)16-s + (0.336 + 0.941i)17-s + (−0.755 − 0.654i)18-s + (0.610 − 0.791i)19-s + (−0.587 + 0.809i)22-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $-0.825 + 0.564i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ -0.825 + 0.564i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6097546582 + 1.970783399i\)
\(L(\frac12)\) \(\approx\) \(0.6097546582 + 1.970783399i\)
\(L(1)\) \(\approx\) \(0.8387805649 + 0.8863646433i\)
\(L(1)\) \(\approx\) \(0.8387805649 + 0.8863646433i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.170 + 0.985i)T \)
3 \( 1 + (0.491 + 0.870i)T \)
11 \( 1 + (0.897 + 0.441i)T \)
13 \( 1 + (0.967 + 0.254i)T \)
17 \( 1 + (0.336 + 0.941i)T \)
19 \( 1 + (0.610 - 0.791i)T \)
29 \( 1 + (-0.610 - 0.791i)T \)
31 \( 1 + (0.870 + 0.491i)T \)
37 \( 1 + (-0.856 - 0.516i)T \)
41 \( 1 + (0.921 + 0.389i)T \)
43 \( 1 + (0.909 + 0.415i)T \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (0.0570 - 0.998i)T \)
59 \( 1 + (-0.254 + 0.967i)T \)
61 \( 1 + (-0.993 + 0.113i)T \)
67 \( 1 + (0.717 - 0.696i)T \)
71 \( 1 + (-0.466 + 0.884i)T \)
73 \( 1 + (0.999 + 0.0285i)T \)
79 \( 1 + (0.362 + 0.931i)T \)
83 \( 1 + (0.226 - 0.974i)T \)
89 \( 1 + (-0.736 - 0.676i)T \)
97 \( 1 + (0.226 + 0.974i)T \)
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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

−18.45398960502565053921766242248, −17.778608771382326953359561271613, −17.09605076698098658836754400912, −16.34212294605523417850736563744, −15.38464638203454081097359063223, −14.316064375732246230462150522872, −13.93249886259932859357757099355, −13.515696761258504763773204061273, −12.43239734878384341099466757557, −12.22411209973237601214623867655, −11.3454832365046854381122902913, −10.840452863346321549774327799610, −9.737219736599325722587416851186, −9.1824405611853942306778935560, −8.58755074149299380953682374137, −7.84688707903061304098290952281, −7.20845601100095165338253628081, −6.14192412486861401454988925653, −5.54870048269208224751229584778, −4.366613687442787056361597950514, −3.48410923204426616960007987220, −3.12442876807466036772651739364, −2.12542281260984171323517928237, −1.26503822220933974176431212120, −0.77767211298501238041872101665, 0.969056804528951470086173474541, 1.98032560537694173277952587765, 3.251537845722209161386853769049, 3.985327771060875150802818535979, 4.43368581381344384047543563735, 5.389046770509033036375588008970, 6.04758591932969148797580387598, 6.82677478497526152399474879597, 7.675249292355257767899194327754, 8.371202613142733900887634657148, 9.04627946865856318006322080293, 9.47171838584357523802682893289, 10.27389857375784416156754494215, 10.94962769002885144861430449205, 11.835145941443518331716256113418, 12.863459895681328465966380200312, 13.62520037257396612952497752142, 14.16740824501281328975681932295, 14.75863629694273446631019142869, 15.50334167815149095687763260538, 15.83602781014622588792003309904, 16.687709791401437948706917288782, 17.18955215936295854432771617440, 17.84416877478689973873416319802, 18.71324484966876774393311437386

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