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Properties

Label 1-4025-4025.48-r0-0-0
Degree $1$
Conductor $4025$
Sign $0.520 + 0.853i$
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  + (−0.113 − 0.993i)2-s + (0.336 − 0.941i)3-s + (−0.974 + 0.226i)4-s + (−0.974 − 0.226i)6-s + (0.336 + 0.941i)8-s + (−0.774 − 0.633i)9-s + (−0.736 − 0.676i)11-s + (−0.113 + 0.993i)12-s + (0.170 + 0.985i)13-s + (0.897 − 0.441i)16-s + (−0.226 + 0.974i)17-s + (−0.540 + 0.841i)18-s + (0.0855 − 0.996i)19-s + (−0.587 + 0.809i)22-s + 24-s + ⋯
L(s)  = 1  + (−0.113 − 0.993i)2-s + (0.336 − 0.941i)3-s + (−0.974 + 0.226i)4-s + (−0.974 − 0.226i)6-s + (0.336 + 0.941i)8-s + (−0.774 − 0.633i)9-s + (−0.736 − 0.676i)11-s + (−0.113 + 0.993i)12-s + (0.170 + 0.985i)13-s + (0.897 − 0.441i)16-s + (−0.226 + 0.974i)17-s + (−0.540 + 0.841i)18-s + (0.0855 − 0.996i)19-s + (−0.587 + 0.809i)22-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $0.520 + 0.853i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ 0.520 + 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1808257440 - 0.1014964298i\)
\(L(\frac12)\) \(\approx\) \(-0.1808257440 - 0.1014964298i\)
\(L(1)\) \(\approx\) \(0.5270699557 - 0.5859223587i\)
\(L(1)\) \(\approx\) \(0.5270699557 - 0.5859223587i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.113 - 0.993i)T \)
3 \( 1 + (0.336 - 0.941i)T \)
11 \( 1 + (-0.736 - 0.676i)T \)
13 \( 1 + (0.170 + 0.985i)T \)
17 \( 1 + (-0.226 + 0.974i)T \)
19 \( 1 + (0.0855 - 0.996i)T \)
29 \( 1 + (-0.0855 - 0.996i)T \)
31 \( 1 + (-0.941 + 0.336i)T \)
37 \( 1 + (0.633 - 0.774i)T \)
41 \( 1 + (0.254 - 0.967i)T \)
43 \( 1 + (-0.281 - 0.959i)T \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (-0.884 - 0.466i)T \)
59 \( 1 + (-0.985 + 0.170i)T \)
61 \( 1 + (0.564 - 0.825i)T \)
67 \( 1 + (-0.491 + 0.870i)T \)
71 \( 1 + (0.198 - 0.980i)T \)
73 \( 1 + (0.856 - 0.516i)T \)
79 \( 1 + (-0.696 + 0.717i)T \)
83 \( 1 + (0.931 + 0.362i)T \)
89 \( 1 + (-0.0285 + 0.999i)T \)
97 \( 1 + (0.931 - 0.362i)T \)
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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

−18.73710074638083719064013590942, −18.2603407340049503048553097953, −17.60422048192287463068614133958, −16.788226200091863194859689621488, −16.16364446536599883222470225828, −15.740058205398227793295791356598, −14.96993975911945982610895069655, −14.60026435895078549071997516474, −13.80609328405904448500863910506, −13.065676770810626637642897379253, −12.45963881224049531768334245903, −11.23463888040035701770653910700, −10.53260115455414691045887926831, −9.84235682395337688398997612036, −9.40020026093651712836364805194, −8.54516896106649275684851628585, −7.80702374965876063218832135411, −7.451326823124131684891893934415, −6.260949352802934685728336277860, −5.5601835715080814929629374976, −4.91597675397456365321440346631, −4.36310623240114617919393178713, −3.36466091702543818315112714738, −2.73243091685107433353451630827, −1.38637937938939502417767240510, 0.06446574148368448120312973436, 1.00499927526799555064708539433, 2.073348038108857639366630363914, 2.352405021046555258884376882603, 3.44114306205248531941235556957, 3.98073888689836627914866397532, 5.08500013033043942676546185358, 5.856501525755375816803334230053, 6.70977237386405096578052253914, 7.60445852259070435734275511096, 8.2401985731508680267467531947, 8.96949933323833218176617562953, 9.37210352812462866263373599989, 10.57982978977971609211108851888, 11.04454426964068539832426336103, 11.75095930360426799280759377011, 12.4335027608575989894554793273, 13.132633809015209764007657954622, 13.57751396971393495935215621662, 14.17034955668063522638736994331, 14.95345394235755028849718109360, 15.87973891295900127803628759299, 16.90710151575370914489488057241, 17.39197749488459173988075651896, 18.2048971059287318005949902766

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