↑

Properties

Label 1-35e2-1225.554-r0-0-0
Degree $1$
Conductor $1225$
Sign $0.873 + 0.486i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.473 − 0.880i)2-s + (−0.858 + 0.512i)3-s + (−0.550 + 0.834i)4-s + (0.858 + 0.512i)6-s + (0.995 + 0.0896i)8-s + (0.473 − 0.880i)9-s + (0.473 + 0.880i)11-s + (0.0448 − 0.998i)12-s + (−0.983 + 0.178i)13-s + (−0.393 − 0.919i)16-s + (0.550 + 0.834i)17-s − 18-s + (0.309 − 0.951i)19-s + (0.550 − 0.834i)22-s + (0.0448 + 0.998i)23-s + (−0.900 + 0.433i)24-s + ⋯
L(s)  = 1  + (−0.473 − 0.880i)2-s + (−0.858 + 0.512i)3-s + (−0.550 + 0.834i)4-s + (0.858 + 0.512i)6-s + (0.995 + 0.0896i)8-s + (0.473 − 0.880i)9-s + (0.473 + 0.880i)11-s + (0.0448 − 0.998i)12-s + (−0.983 + 0.178i)13-s + (−0.393 − 0.919i)16-s + (0.550 + 0.834i)17-s − 18-s + (0.309 − 0.951i)19-s + (0.550 − 0.834i)22-s + (0.0448 + 0.998i)23-s + (−0.900 + 0.433i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.873 + 0.486i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (554, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ 0.873 + 0.486i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7201563023 + 0.1868746544i\)
\(L(\frac12)\) \(\approx\) \(0.7201563023 + 0.1868746544i\)
\(L(1)\) \(\approx\) \(0.6388628907 - 0.04775560060i\)
\(L(1)\) \(\approx\) \(0.6388628907 - 0.04775560060i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.473 - 0.880i)T \)
3 \( 1 + (-0.858 + 0.512i)T \)
11 \( 1 + (0.473 + 0.880i)T \)
13 \( 1 + (-0.983 + 0.178i)T \)
17 \( 1 + (0.550 + 0.834i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.0448 + 0.998i)T \)
29 \( 1 + (0.936 + 0.351i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.0448 - 0.998i)T \)
41 \( 1 + (0.753 + 0.657i)T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (-0.134 + 0.990i)T \)
53 \( 1 + (0.550 - 0.834i)T \)
59 \( 1 + (-0.393 - 0.919i)T \)
61 \( 1 + (-0.963 + 0.266i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (-0.550 + 0.834i)T \)
73 \( 1 + (-0.983 - 0.178i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.134 - 0.990i)T \)
89 \( 1 + (0.983 + 0.178i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.25213785879148387809307615456, −19.96008864051191684031231935609, −19.22849860065787306286270993525, −18.60092580764578320444530013690, −17.95267832752859889527221103415, −17.14104092277085147932104065251, −16.49953446589294208738155005211, −16.10004227473798917300566614484, −14.94606281080183696525865928710, −14.09609831641488691506489891216, −13.57756978125165082280159799036, −12.33673901589844541617914058145, −11.84217704815338551019107095984, −10.64869802743847399876063716611, −10.13233590615039086977373961714, −9.12285093781707857975699186568, −8.121985186020184154266619084593, −7.48805741572581208969363393960, −6.60815848490087460547810677567, −5.9779091749996608915508068416, −5.141052752233222959258558492862, −4.40973147539580263608905762559, −2.85564648598735418999386428952, −1.43804409832250506274787170879, −0.557783363603309715625308552964, 0.90507164598068318979440848315, 1.970986834288229632808164313046, 3.11924098870272082019822371797, 4.17833392805135202351544110825, 4.718245541906438674210043355437, 5.744632803047681488678223852174, 6.9775736412699361272133048628, 7.61909798491594742945894260045, 8.92280731601350503354040980051, 9.65092626967502618607959765028, 10.11327491006957864800772099959, 11.053795768160021280840719955651, 11.74629619897485559748783933409, 12.36798304268460492043055785549, 13.03063659756816090230138280014, 14.24941628573640931907355570548, 15.097771696142217043913183854687, 16.01788751755624786533752563607, 16.8969725807929553724761711649, 17.55034056197508091574763490889, 17.818113654569489787541117818345, 19.05746371910202250555495278706, 19.64479765961610813512466040230, 20.42350450396327427230693103410, 21.33495657319088900775203106712

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