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Properties

Label 1-4025-4025.206-r0-0-0
Degree $1$
Conductor $4025$
Sign $-0.957 - 0.290i$
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  + (−0.104 − 0.994i)2-s + (0.978 − 0.207i)3-s + (−0.978 + 0.207i)4-s + (−0.309 − 0.951i)6-s + (0.309 + 0.951i)8-s + (0.913 − 0.406i)9-s + (−0.913 − 0.406i)11-s + (−0.913 + 0.406i)12-s + (0.809 + 0.587i)13-s + (0.913 − 0.406i)16-s + (0.669 − 0.743i)17-s + (−0.5 − 0.866i)18-s + (−0.978 − 0.207i)19-s + (−0.309 + 0.951i)22-s + (0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (0.978 − 0.207i)3-s + (−0.978 + 0.207i)4-s + (−0.309 − 0.951i)6-s + (0.309 + 0.951i)8-s + (0.913 − 0.406i)9-s + (−0.913 − 0.406i)11-s + (−0.913 + 0.406i)12-s + (0.809 + 0.587i)13-s + (0.913 − 0.406i)16-s + (0.669 − 0.743i)17-s + (−0.5 − 0.866i)18-s + (−0.978 − 0.207i)19-s + (−0.309 + 0.951i)22-s + (0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $-0.957 - 0.290i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ -0.957 - 0.290i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2461539722 - 1.660707041i\)
\(L(\frac12)\) \(\approx\) \(0.2461539722 - 1.660707041i\)
\(L(1)\) \(\approx\) \(0.9420663498 - 0.7391011779i\)
\(L(1)\) \(\approx\) \(0.9420663498 - 0.7391011779i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.104 - 0.994i)T \)
3 \( 1 + (0.978 - 0.207i)T \)
11 \( 1 + (-0.913 - 0.406i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 + (0.669 - 0.743i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (-0.913 + 0.406i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.669 - 0.743i)T \)
53 \( 1 + (0.978 - 0.207i)T \)
59 \( 1 + (0.104 - 0.994i)T \)
61 \( 1 + (-0.104 - 0.994i)T \)
67 \( 1 + (-0.669 + 0.743i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.913 - 0.406i)T \)
79 \( 1 + (-0.669 - 0.743i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (-0.104 - 0.994i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−18.61229536823913990959866052907, −18.18498914551275365196565806221, −17.397484251086040975278337384253, −16.516099153408311813494041060628, −15.98581341981809585933340261463, −15.29212891063713586670897174432, −14.82811547816647683598864991380, −14.25463118248093752764418737389, −13.34006788164037058239060836984, −12.97245915888695149230189827216, −12.338172969599233072777366723236, −10.73305754948209247468528382399, −10.39322337180843236098223289305, −9.67492758483732889965441291417, −8.645259062503599605631946420248, −8.48633854009971039659655599533, −7.60462553358557965416290731121, −7.14561075410118906859550168767, −6.07027052491840799406622517139, −5.45771432451603946711103008439, −4.5657848588385902841700924582, −3.84369461825591151486865235582, −3.17200286721048971991247038205, −2.08063659162565826463812175536, −1.146567223407554435114569394207, 0.43889714213517209651879521526, 1.54413654934631547458867306774, 2.16439394811787142019903661655, 3.04908417540012546714562278441, 3.506768526832515816824307625025, 4.41234154245015830874837487721, 5.12101597480290310529800386295, 6.1890451189745987040691605304, 7.15475113403137959657669528993, 8.011778908139694546626534834110, 8.52960994085128523197933257426, 9.0921181494486267971790113511, 9.955038184041552941369580515638, 10.44428589223181308104092644322, 11.31586569713778489682717935611, 11.97015351129103279808258651843, 12.86491133980328781894892130884, 13.29086518875519136349749317757, 13.914679001892921212642688484774, 14.48225488081439512271735430949, 15.34466076539483952695579802820, 16.12162813469470243107728864191, 16.84268308996206728668283041108, 17.95271577136805654919467207310, 18.31899858187830054998029768607

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