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Properties

Label 1-65e2-4225.1264-r0-0-0
Degree $1$
Conductor $4225$
Sign $-0.921 + 0.388i$
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  + (−0.991 + 0.128i)2-s + (−0.619 + 0.784i)3-s + (0.966 − 0.254i)4-s + (0.513 − 0.857i)6-s + (−0.278 − 0.960i)7-s + (−0.926 + 0.377i)8-s + (−0.231 − 0.972i)9-s + (−0.231 + 0.972i)11-s + (−0.399 + 0.916i)12-s + (0.399 + 0.916i)14-s + (0.870 − 0.493i)16-s + (0.789 + 0.613i)17-s + (0.354 + 0.935i)18-s + (0.913 + 0.406i)19-s + (0.926 + 0.377i)21-s + (0.104 − 0.994i)22-s + ⋯
L(s)  = 1  + (−0.991 + 0.128i)2-s + (−0.619 + 0.784i)3-s + (0.966 − 0.254i)4-s + (0.513 − 0.857i)6-s + (−0.278 − 0.960i)7-s + (−0.926 + 0.377i)8-s + (−0.231 − 0.972i)9-s + (−0.231 + 0.972i)11-s + (−0.399 + 0.916i)12-s + (0.399 + 0.916i)14-s + (0.870 − 0.493i)16-s + (0.789 + 0.613i)17-s + (0.354 + 0.935i)18-s + (0.913 + 0.406i)19-s + (0.926 + 0.377i)21-s + (0.104 − 0.994i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $-0.921 + 0.388i$
Analytic conductor: \(19.6208\)
Root analytic conductor: \(19.6208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4225} (1264, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4225,\ (0:\ ),\ -0.921 + 0.388i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06770758307 + 0.3352560807i\)
\(L(\frac12)\) \(\approx\) \(0.06770758307 + 0.3352560807i\)
\(L(1)\) \(\approx\) \(0.4885711872 + 0.1380519056i\)
\(L(1)\) \(\approx\) \(0.4885711872 + 0.1380519056i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.991 + 0.128i)T \)
3 \( 1 + (-0.619 + 0.784i)T \)
7 \( 1 + (-0.278 - 0.960i)T \)
11 \( 1 + (-0.231 + 0.972i)T \)
17 \( 1 + (0.789 + 0.613i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.669 - 0.743i)T \)
29 \( 1 + (0.184 - 0.982i)T \)
31 \( 1 + (0.485 + 0.873i)T \)
37 \( 1 + (0.324 - 0.945i)T \)
41 \( 1 + (-0.962 + 0.270i)T \)
43 \( 1 + (-0.799 + 0.600i)T \)
47 \( 1 + (-0.836 + 0.548i)T \)
53 \( 1 + (-0.644 - 0.764i)T \)
59 \( 1 + (-0.993 + 0.112i)T \)
61 \( 1 + (0.899 - 0.435i)T \)
67 \( 1 + (-0.966 - 0.254i)T \)
71 \( 1 + (0.369 + 0.929i)T \)
73 \( 1 + (-0.958 + 0.285i)T \)
79 \( 1 + (0.836 - 0.548i)T \)
83 \( 1 + (0.989 - 0.144i)T \)
89 \( 1 + (0.669 + 0.743i)T \)
97 \( 1 + (0.581 + 0.813i)T \)
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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

−18.28045252299813512100709303648, −17.58523874143514100584019774327, −16.74574793228742572263866706768, −16.26306710317602743747964087596, −15.71370986753747905287822091930, −14.89044694270894176206927585596, −13.775026791409407148508613128681, −13.28427293322719479103852216486, −12.248922797773612371931025629412, −11.81908400453006248586457634378, −11.42690405726591357583862121175, −10.52431959211756165945865618407, −9.79587952240377401221757278693, −9.075902097129377104845855942822, −8.26398209561050508210545940146, −7.79007661046825823336596721394, −6.96435968126258142295812242746, −6.25374278504910150518391451108, −5.62946328846136611570592647021, −5.031663599823656122278492104896, −3.293524837730564138229156009351, −2.92304363642987283153037103513, −1.91567007070802988710531077557, −1.16743657025186135485240710785, −0.184196273801999237381431196529, 0.916852833911059466803348460124, 1.750343773284004762042340321799, 2.98125939982015429915736026786, 3.69393624709108489021318719827, 4.58942264701954742748656548523, 5.35706062133348124399750678582, 6.3077305150892076672242078294, 6.72180252715381221615417689685, 7.73112208127532500826386361966, 8.14924189822110248186106596772, 9.34005667325526796212304426291, 9.86224884612136679035326897730, 10.275145581724027531100672352130, 10.80672486825891736778048800081, 11.77994349951043404830032873131, 12.19055965060399405421700519750, 13.090788460453523903033776424151, 14.31401941414379238982614753774, 14.71331882038900726027941248482, 15.594000978589600602209435622089, 16.22424608774047445929717300780, 16.56726902493838512954334578518, 17.43176445762579241427934335961, 17.71703735321015108627613011598, 18.48599857377464681573749179413

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