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Properties

Label 1-35e2-1225.1066-r0-0-0
Degree $1$
Conductor $1225$
Sign $0.475 + 0.879i$
Analytic cond. $5.68887$
Root an. cond. $5.68887$
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  + (0.998 + 0.0598i)2-s + (0.525 + 0.850i)3-s + (0.992 + 0.119i)4-s + (0.473 + 0.880i)6-s + (0.983 + 0.178i)8-s + (−0.447 + 0.894i)9-s + (−0.447 − 0.894i)11-s + (0.420 + 0.907i)12-s + (0.936 − 0.351i)13-s + (0.971 + 0.237i)16-s + (−0.599 − 0.800i)17-s + (−0.5 + 0.866i)18-s + (−0.104 + 0.994i)19-s + (−0.393 − 0.919i)22-s + (0.575 + 0.817i)23-s + (0.365 + 0.930i)24-s + ⋯
L(s)  = 1  + (0.998 + 0.0598i)2-s + (0.525 + 0.850i)3-s + (0.992 + 0.119i)4-s + (0.473 + 0.880i)6-s + (0.983 + 0.178i)8-s + (−0.447 + 0.894i)9-s + (−0.447 − 0.894i)11-s + (0.420 + 0.907i)12-s + (0.936 − 0.351i)13-s + (0.971 + 0.237i)16-s + (−0.599 − 0.800i)17-s + (−0.5 + 0.866i)18-s + (−0.104 + 0.994i)19-s + (−0.393 − 0.919i)22-s + (0.575 + 0.817i)23-s + (0.365 + 0.930i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.475 + 0.879i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (1066, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ 0.475 + 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.221924237 + 1.921274543i\)
\(L(\frac12)\) \(\approx\) \(3.221924237 + 1.921274543i\)
\(L(1)\) \(\approx\) \(2.224745002 + 0.7656624827i\)
\(L(1)\) \(\approx\) \(2.224745002 + 0.7656624827i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.998 + 0.0598i)T \)
3 \( 1 + (0.525 + 0.850i)T \)
11 \( 1 + (-0.447 - 0.894i)T \)
13 \( 1 + (0.936 - 0.351i)T \)
17 \( 1 + (-0.599 - 0.800i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (0.575 + 0.817i)T \)
29 \( 1 + (0.753 + 0.657i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (0.420 + 0.907i)T \)
41 \( 1 + (0.134 + 0.990i)T \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (0.251 + 0.967i)T \)
53 \( 1 + (0.992 + 0.119i)T \)
59 \( 1 + (-0.280 - 0.959i)T \)
61 \( 1 + (-0.873 - 0.486i)T \)
67 \( 1 + (0.913 - 0.406i)T \)
71 \( 1 + (-0.393 - 0.919i)T \)
73 \( 1 + (-0.772 + 0.635i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (-0.963 + 0.266i)T \)
89 \( 1 + (-0.163 - 0.986i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−21.04515977179447630848189582451, −20.2422765705211297318304274315, −19.657970478821387041027741881515, −18.91824962518269345210556041975, −17.959990237209977241268449507037, −17.27130511015027350693155979133, −16.13397212197015427100389298689, −15.24187085662654474680150793632, −14.86424931717338091653769156877, −13.73078873926702749492911870218, −13.36647000154389377320708124339, −12.60327447590421635706477043819, −11.93098506320465327362354385078, −11.02043861152153224324494795101, −10.20572203294127337835623396466, −8.91549923824382305919399277452, −8.20002965353416553918299234429, −7.114673499531283877836661436710, −6.65462298791086811482415045074, −5.801510232934465522856173411758, −4.62812486404344868694517150878, −3.89986656063811530793679264429, −2.72814468268797603404377341860, −2.17241830532997854359488096257, −1.09459921594661274327075570480, 1.40621254080591946231254023032, 2.79296988397088599246145043172, 3.21597478338229686440969945931, 4.157145726544795848992197706167, 5.00885150936752640012413438878, 5.75468697298842169430390127465, 6.63061460742837407799715631429, 7.91312750805153140433495499704, 8.38398965268594491580766103090, 9.53541189668857467574164370257, 10.49848343932505504969455136190, 11.09414936705518504700463494759, 11.81760331055488323849542065090, 13.07085603440007999826571057485, 13.62732225647508477687888806081, 14.185010212993555747638397067500, 15.18933110113601065603791032353, 15.70263184821153532153925355364, 16.31981386872001749281436640459, 17.04474124548609659118704427902, 18.351589996150336337146360294837, 19.20175874249383220502341883143, 20.113218878244056845906625369455, 20.64056827515424035246259122893, 21.36023920335724851578272190608

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