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Properties

Label 1-65e2-4225.614-r0-0-0
Degree $1$
Conductor $4225$
Sign $-0.694 - 0.719i$
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  + (0.932 − 0.362i)2-s + (−0.853 − 0.520i)3-s + (0.737 − 0.675i)4-s + (−0.984 − 0.176i)6-s + (0.845 + 0.534i)7-s + (0.443 − 0.896i)8-s + (0.457 + 0.889i)9-s + (0.457 − 0.889i)11-s + (−0.981 + 0.192i)12-s + (0.981 + 0.192i)14-s + (0.0884 − 0.996i)16-s + (−0.997 + 0.0643i)17-s + (0.748 + 0.663i)18-s + (0.913 + 0.406i)19-s + (−0.443 − 0.896i)21-s + (0.104 − 0.994i)22-s + ⋯
L(s)  = 1  + (0.932 − 0.362i)2-s + (−0.853 − 0.520i)3-s + (0.737 − 0.675i)4-s + (−0.984 − 0.176i)6-s + (0.845 + 0.534i)7-s + (0.443 − 0.896i)8-s + (0.457 + 0.889i)9-s + (0.457 − 0.889i)11-s + (−0.981 + 0.192i)12-s + (0.981 + 0.192i)14-s + (0.0884 − 0.996i)16-s + (−0.997 + 0.0643i)17-s + (0.748 + 0.663i)18-s + (0.913 + 0.406i)19-s + (−0.443 − 0.896i)21-s + (0.104 − 0.994i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $-0.694 - 0.719i$
Analytic conductor: \(19.6208\)
Root analytic conductor: \(19.6208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4225} (614, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4225,\ (0:\ ),\ -0.694 - 0.719i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9555005668 - 2.248822508i\)
\(L(\frac12)\) \(\approx\) \(0.9555005668 - 2.248822508i\)
\(L(1)\) \(\approx\) \(1.332713302 - 0.7850582471i\)
\(L(1)\) \(\approx\) \(1.332713302 - 0.7850582471i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.932 - 0.362i)T \)
3 \( 1 + (-0.853 - 0.520i)T \)
7 \( 1 + (0.845 + 0.534i)T \)
11 \( 1 + (0.457 - 0.889i)T \)
17 \( 1 + (-0.997 + 0.0643i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.669 - 0.743i)T \)
29 \( 1 + (0.0563 + 0.998i)T \)
31 \( 1 + (0.644 - 0.764i)T \)
37 \( 1 + (-0.999 + 0.0322i)T \)
41 \( 1 + (-0.384 - 0.923i)T \)
43 \( 1 + (-0.278 - 0.960i)T \)
47 \( 1 + (-0.995 + 0.0965i)T \)
53 \( 1 + (0.989 + 0.144i)T \)
59 \( 1 + (-0.657 - 0.753i)T \)
61 \( 1 + (-0.962 - 0.270i)T \)
67 \( 1 + (-0.737 - 0.675i)T \)
71 \( 1 + (-0.877 + 0.478i)T \)
73 \( 1 + (0.998 - 0.0483i)T \)
79 \( 1 + (0.995 - 0.0965i)T \)
83 \( 1 + (-0.0241 - 0.999i)T \)
89 \( 1 + (0.669 + 0.743i)T \)
97 \( 1 + (0.999 - 0.0161i)T \)
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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

−18.11077954600006433439036256804, −17.73424461622668971025739025404, −17.26916749273683432671743702675, −16.50481677984061078499641319771, −15.84490253246493546533762162916, −15.19704788181244469295740217487, −14.76930734776321902980449119580, −13.77885439316741641771020301662, −13.40244283014930758537404712208, −12.34716621455712268710973190268, −11.70362719500835680599421503601, −11.44884084465546875720841383835, −10.548501657904468698220783955846, −9.89523694805416363716624360475, −8.96965665193066523399402345803, −7.96344559590300957974980898759, −7.24346344803478350714455459373, −6.6513412098335799380870132220, −5.941567508298116238895863618565, −4.95511714292189338022790057057, −4.68858626121434619120958299915, −4.01050598179383819206766803111, −3.19154846479764545499553853630, −1.996109096225585668638021424620, −1.21897306195880816019768116401, 0.52985837806095146463407500464, 1.59148174585880518702638597106, 2.0505171949283312048145062151, 3.09105623125121220729602760418, 4.02443035063933271123121006706, 4.84831191970398335162199551944, 5.3684003789880558365433221859, 6.10174962581607905683954735462, 6.62905277817066752956401322698, 7.48584962818524685982168555976, 8.317059009811090139470567780748, 9.15593524289620750080323845813, 10.33488273103673427443891439286, 10.80441694359336727875700616912, 11.51379572265641260350255341370, 11.99530991966975972824985805161, 12.439570015203797817364655883914, 13.5001735563221447783322891795, 13.829815184882031717917222023952, 14.55842628790945546553036027824, 15.43419198025147506545608972024, 16.01402143549073319922108440758, 16.71029974860003280388208454762, 17.46232005914917326972473446658, 18.334864148678793078929280258616

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