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Properties

Label 1-4025-4025.1028-r0-0-0
Degree $1$
Conductor $4025$
Sign $-0.153 + 0.988i$
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  + (0.884 + 0.466i)2-s + (0.113 − 0.993i)3-s + (0.564 + 0.825i)4-s + (0.564 − 0.825i)6-s + (0.113 + 0.993i)8-s + (−0.974 − 0.226i)9-s + (0.696 − 0.717i)11-s + (0.884 − 0.466i)12-s + (0.0570 + 0.998i)13-s + (−0.362 + 0.931i)16-s + (−0.825 − 0.564i)17-s + (−0.755 − 0.654i)18-s + (−0.0285 + 0.999i)19-s + (0.951 − 0.309i)22-s + 24-s + ⋯
L(s)  = 1  + (0.884 + 0.466i)2-s + (0.113 − 0.993i)3-s + (0.564 + 0.825i)4-s + (0.564 − 0.825i)6-s + (0.113 + 0.993i)8-s + (−0.974 − 0.226i)9-s + (0.696 − 0.717i)11-s + (0.884 − 0.466i)12-s + (0.0570 + 0.998i)13-s + (−0.362 + 0.931i)16-s + (−0.825 − 0.564i)17-s + (−0.755 − 0.654i)18-s + (−0.0285 + 0.999i)19-s + (0.951 − 0.309i)22-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $-0.153 + 0.988i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (1028, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ -0.153 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.454407127 + 1.697370559i\)
\(L(\frac12)\) \(\approx\) \(1.454407127 + 1.697370559i\)
\(L(1)\) \(\approx\) \(1.585426992 + 0.3150681360i\)
\(L(1)\) \(\approx\) \(1.585426992 + 0.3150681360i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.884 + 0.466i)T \)
3 \( 1 + (0.113 - 0.993i)T \)
11 \( 1 + (0.696 - 0.717i)T \)
13 \( 1 + (0.0570 + 0.998i)T \)
17 \( 1 + (-0.825 - 0.564i)T \)
19 \( 1 + (-0.0285 + 0.999i)T \)
29 \( 1 + (0.0285 + 0.999i)T \)
31 \( 1 + (-0.993 + 0.113i)T \)
37 \( 1 + (0.226 - 0.974i)T \)
41 \( 1 + (-0.0855 + 0.996i)T \)
43 \( 1 + (0.909 + 0.415i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (-0.633 + 0.774i)T \)
59 \( 1 + (-0.998 + 0.0570i)T \)
61 \( 1 + (-0.198 + 0.980i)T \)
67 \( 1 + (-0.170 + 0.985i)T \)
71 \( 1 + (0.897 - 0.441i)T \)
73 \( 1 + (0.336 - 0.941i)T \)
79 \( 1 + (0.254 - 0.967i)T \)
83 \( 1 + (0.389 + 0.921i)T \)
89 \( 1 + (-0.870 + 0.491i)T \)
97 \( 1 + (0.389 - 0.921i)T \)
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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

−18.32330241198423060256639636017, −17.38040762828534174152936905150, −16.98067292467186099900065619231, −15.80187927859178225308888247241, −15.45544384435840575028284872827, −14.96104397492109046028009354431, −14.26570909564858965478218224547, −13.52379138465521826501705698478, −12.83469630249596977528225997214, −12.141527529203258300239871531486, −11.1997601930052237890704174771, −10.94640121883718213730099184685, −10.035082839652483886545639866605, −9.53552412936617820915067936212, −8.773992706318106822494279793177, −7.806142057458526083807362394034, −6.78749041352367445245152802483, −6.12235039856224933316154963695, −5.27348339449405980022633951866, −4.67373363992256803238796207682, −4.00499545022168349118490129066, −3.362078329847049341953074496214, −2.50995257087694728936585239410, −1.78194293848917348038725276544, −0.40440350312763936427432637041, 1.290944562390256682588777859056, 1.98906234447025643321795193525, 2.90024844697516742563661690276, 3.63491259627991869386393935065, 4.42019097615118331798984472749, 5.36244145859084391374083847343, 6.2012673047277844662912937242, 6.531692337954140448466255795077, 7.35345749813425497921985673906, 7.924260435945292065844890632751, 8.887290877924361496254445039196, 9.222367409535389733884521108122, 10.84876569657345341870560748161, 11.31887364229785227178116929905, 11.99097515132831144526846276928, 12.6059405757049218365402972670, 13.26068553048382073203871373088, 14.03183885462190765655344192785, 14.313426191423064376883085663749, 14.96022264705398145673862016812, 16.176776979786714226639607778477, 16.43489670583927604505567443770, 17.16677866901472742699161234728, 18.0268667089670463367323354270, 18.512166829494378263332493256017

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