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Properties

Label 1-1925-1925.1209-r0-0-0
Degree $1$
Conductor $1925$
Sign $0.968 - 0.249i$
Analytic cond. $8.93966$
Root an. cond. $8.93966$
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  + (0.669 − 0.743i)2-s + (−0.104 − 0.994i)3-s + (−0.104 − 0.994i)4-s + (−0.809 − 0.587i)6-s + (−0.809 − 0.587i)8-s + (−0.978 + 0.207i)9-s + (−0.978 + 0.207i)12-s + (−0.309 + 0.951i)13-s + (−0.978 + 0.207i)16-s + (−0.913 + 0.406i)17-s + (−0.5 + 0.866i)18-s + (−0.104 + 0.994i)19-s + (−0.669 + 0.743i)23-s + (−0.5 + 0.866i)24-s + (0.5 + 0.866i)26-s + (0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)2-s + (−0.104 − 0.994i)3-s + (−0.104 − 0.994i)4-s + (−0.809 − 0.587i)6-s + (−0.809 − 0.587i)8-s + (−0.978 + 0.207i)9-s + (−0.978 + 0.207i)12-s + (−0.309 + 0.951i)13-s + (−0.978 + 0.207i)16-s + (−0.913 + 0.406i)17-s + (−0.5 + 0.866i)18-s + (−0.104 + 0.994i)19-s + (−0.669 + 0.743i)23-s + (−0.5 + 0.866i)24-s + (0.5 + 0.866i)26-s + (0.309 + 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.968 - 0.249i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ 0.968 - 0.249i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.135936705 - 0.1441292801i\)
\(L(\frac12)\) \(\approx\) \(1.135936705 - 0.1441292801i\)
\(L(1)\) \(\approx\) \(0.9298855099 - 0.6320669604i\)
\(L(1)\) \(\approx\) \(0.9298855099 - 0.6320669604i\)

Euler product

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

−20.163423146544438074839491986504, −19.762736724796659932887969397833, −18.11044451776500422945705482949, −17.802688426349101836584688594921, −16.898352936218947457910500588544, −16.24832723946095528509739367630, −15.63189930036430786421691649460, −15.00497136786095133682361008387, −14.43463264803308179905046869037, −13.543614647977944067626735636638, −12.83860417838333068619369213142, −11.96016867070506819107814484534, −11.16602342706002358640296749187, −10.45534753558266037001080523249, −9.409266311490388412838297104280, −8.772980360003467666976076934517, −7.98297113108328725325787799731, −7.05188876326056840764067888245, −6.16491048033364946099827522257, −5.45376107975169463175697254361, −4.62003841860536973103036847125, −4.16278795998638194728452368372, −3.004492482500469738049217784420, −2.503723522970469940925231071988, −0.32712170556003237152111526534, 1.11703475005464983567953268996, 1.98559352989624974658003286412, 2.53557501772705319792824136064, 3.75506799809106306853450272547, 4.433247226583954314517469402139, 5.58938427019387529171028983252, 6.13755114639106663732231105224, 6.93903492697147748626718441154, 7.8214544387255419217500611286, 8.860432917843485880985433951791, 9.55425050926846331557342810028, 10.641530164090116879856438342002, 11.23919552478255608833345287903, 12.086347707891973620659357004786, 12.468438917701908830874076541539, 13.32222797996222570891065357788, 14.036574109324249355608582685202, 14.43066303494859881524621220509, 15.46068668010612385354297897064, 16.325307330195113834847855110979, 17.35403461441105072142220294015, 17.965082629460828061924213185493, 18.84904478852687434070270888584, 19.25017100796424947955202140250, 19.983112243279262001965777011324

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