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Properties

Label 1-35e2-1225.33-r0-0-0
Degree $1$
Conductor $1225$
Sign $0.932 + 0.360i$
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  + (−0.981 − 0.193i)2-s + (0.986 + 0.163i)3-s + (0.925 + 0.379i)4-s + (−0.936 − 0.351i)6-s + (−0.834 − 0.550i)8-s + (0.946 + 0.323i)9-s + (−0.946 + 0.323i)11-s + (0.850 + 0.525i)12-s + (0.919 + 0.393i)13-s + (0.712 + 0.701i)16-s + (−0.611 + 0.791i)17-s + (−0.866 − 0.5i)18-s + (0.669 − 0.743i)19-s + (0.990 − 0.134i)22-s + (−0.0299 − 0.999i)23-s + (−0.733 − 0.680i)24-s + ⋯
L(s)  = 1  + (−0.981 − 0.193i)2-s + (0.986 + 0.163i)3-s + (0.925 + 0.379i)4-s + (−0.936 − 0.351i)6-s + (−0.834 − 0.550i)8-s + (0.946 + 0.323i)9-s + (−0.946 + 0.323i)11-s + (0.850 + 0.525i)12-s + (0.919 + 0.393i)13-s + (0.712 + 0.701i)16-s + (−0.611 + 0.791i)17-s + (−0.866 − 0.5i)18-s + (0.669 − 0.743i)19-s + (0.990 − 0.134i)22-s + (−0.0299 − 0.999i)23-s + (−0.733 − 0.680i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.932 + 0.360i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ 0.932 + 0.360i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.441643637 + 0.2686271823i\)
\(L(\frac12)\) \(\approx\) \(1.441643637 + 0.2686271823i\)
\(L(1)\) \(\approx\) \(1.030341173 + 0.05713073723i\)
\(L(1)\) \(\approx\) \(1.030341173 + 0.05713073723i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.981 - 0.193i)T \)
3 \( 1 + (0.986 + 0.163i)T \)
11 \( 1 + (-0.946 + 0.323i)T \)
13 \( 1 + (0.919 + 0.393i)T \)
17 \( 1 + (-0.611 + 0.791i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (-0.0299 - 0.999i)T \)
29 \( 1 + (0.691 - 0.722i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (-0.850 - 0.525i)T \)
41 \( 1 + (0.0448 + 0.998i)T \)
43 \( 1 + (0.781 + 0.623i)T \)
47 \( 1 + (-0.907 + 0.420i)T \)
53 \( 1 + (0.379 - 0.925i)T \)
59 \( 1 + (0.251 - 0.967i)T \)
61 \( 1 + (0.646 + 0.762i)T \)
67 \( 1 + (-0.207 + 0.978i)T \)
71 \( 1 + (0.134 + 0.990i)T \)
73 \( 1 + (0.119 - 0.992i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (0.0896 + 0.995i)T \)
89 \( 1 + (-0.599 + 0.800i)T \)
97 \( 1 + (0.951 + 0.309i)T \)
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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

−20.81998915044179615313631699343, −20.26414045234584080253894925772, −19.48652461307495659419439014400, −18.68360413295425370554638368386, −18.21667672131918651936688735708, −17.522415430619576170527904112594, −16.26694140527063035240635372044, −15.68505024582004571048864513481, −15.29512443394529326693427728039, −13.99722688786656267275987985311, −13.6310011329999628372583207856, −12.50154025548013206815166615409, −11.58022505605489047583135467629, −10.57781171608957018077223571310, −9.994205084135577962581232956507, −9.04030160386509582455172805969, −8.44400702029322000369472224361, −7.72022895755682082786690523327, −7.04711360740580006316355257824, −6.01911195324137356426188941718, −5.04821690681788191410526963263, −3.532316983033513246125397693352, −2.865248125717479754383576862719, −1.87365589281658536112233541799, −0.861875812719780432078044482817, 1.04900589289376145023723141321, 2.17387714796077036573046261719, 2.78807413200239562699446601701, 3.81163833126241908705011198408, 4.79972807802606028311932471888, 6.286822812625959774252523561, 6.99605918413138942862855321425, 8.0668664510879368931931422815, 8.41682729317722362238866783745, 9.2818074885097197198676249014, 10.08994144285223108347275010510, 10.71416701337427937170060484451, 11.59392505036522476481107290377, 12.74366091113002071985994724111, 13.29560094058976202601679575713, 14.310735868047468961453427892862, 15.29202109279099387004926981222, 15.81894496036487553640631356590, 16.39813738202818337168331506921, 17.729788058748033403809865583830, 18.05413284051534838420657774175, 19.10201128643661126981934360935, 19.47065161325182272571508788680, 20.41022562880448176962090128036, 20.96115667806117060750158753725

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