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Properties

Label 1-35e2-1225.1053-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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0299 + 0.999i)2-s + (−0.486 + 0.873i)3-s + (−0.998 − 0.0598i)4-s + (−0.858 − 0.512i)6-s + (0.0896 − 0.995i)8-s + (−0.525 − 0.850i)9-s + (0.525 − 0.850i)11-s + (0.538 − 0.842i)12-s + (−0.178 − 0.983i)13-s + (0.992 + 0.119i)16-s + (−0.894 − 0.447i)17-s + (0.866 − 0.5i)18-s + (0.669 + 0.743i)19-s + (0.834 + 0.550i)22-s + (−0.460 + 0.887i)23-s + (0.826 + 0.563i)24-s + ⋯
L(s)  = 1  + (−0.0299 + 0.999i)2-s + (−0.486 + 0.873i)3-s + (−0.998 − 0.0598i)4-s + (−0.858 − 0.512i)6-s + (0.0896 − 0.995i)8-s + (−0.525 − 0.850i)9-s + (0.525 − 0.850i)11-s + (0.538 − 0.842i)12-s + (−0.178 − 0.983i)13-s + (0.992 + 0.119i)16-s + (−0.894 − 0.447i)17-s + (0.866 − 0.5i)18-s + (0.669 + 0.743i)19-s + (0.834 + 0.550i)22-s + (−0.460 + 0.887i)23-s + (0.826 + 0.563i)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: $\chi_{1225} (1053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ -0.873 - 0.486i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1206903738 + 0.4651028445i\)
\(L(\frac12)\) \(\approx\) \(-0.1206903738 + 0.4651028445i\)
\(L(1)\) \(\approx\) \(0.5124496929 + 0.4449903878i\)
\(L(1)\) \(\approx\) \(0.5124496929 + 0.4449903878i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.0299 + 0.999i)T \)
3 \( 1 + (-0.486 + 0.873i)T \)
11 \( 1 + (0.525 - 0.850i)T \)
13 \( 1 + (-0.178 - 0.983i)T \)
17 \( 1 + (-0.894 - 0.447i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (-0.460 + 0.887i)T \)
29 \( 1 + (-0.936 - 0.351i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (-0.538 + 0.842i)T \)
41 \( 1 + (-0.753 - 0.657i)T \)
43 \( 1 + (-0.974 - 0.222i)T \)
47 \( 1 + (-0.611 + 0.791i)T \)
53 \( 1 + (-0.0598 + 0.998i)T \)
59 \( 1 + (-0.599 + 0.800i)T \)
61 \( 1 + (-0.251 + 0.967i)T \)
67 \( 1 + (0.207 + 0.978i)T \)
71 \( 1 + (-0.550 + 0.834i)T \)
73 \( 1 + (-0.941 + 0.337i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (0.990 - 0.134i)T \)
89 \( 1 + (-0.646 + 0.762i)T \)
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

−20.4556435347154728115215196683, −19.82717523291234719255455687668, −19.27760460931635706725889967387, −18.376801993548056943230639076559, −17.85521238577186167837216337800, −17.14527012014913592649534361081, −16.41286507189401953999165917849, −15.05479838505060127985672933882, −14.182089837861856329301571163735, −13.48893916340267463400712929038, −12.75316608548113155931817786140, −12.00698393657601387343229361439, −11.490269966624113877578054300467, −10.70012651038428263588253225275, −9.72421255327795106147033724549, −8.926464438127495085663364237508, −8.069751163312798888567706806185, −6.98026358180698782386522908296, −6.36267438489636955070394577454, −5.0220378157261724109909461833, −4.49862472572470600678360638249, −3.29190795921204122404822281866, −2.02143489031789077294044520561, −1.71933220500842052276794685843, −0.23687021037479720759532155229, 1.09513074607774032864604340053, 3.10236607584589302530874387038, 3.82985553567572855670624227732, 4.76614577695519802276767228150, 5.61993802294673429586349689095, 6.10364245741894238652234003064, 7.14617873922284524222534305248, 8.1472793184202298617869649108, 8.89409691113664894932579644945, 9.728745278688628935214970249240, 10.34641786733071286741007152427, 11.43209695434484148502980211746, 12.136023137238692251185172875225, 13.38610917290032081323396326846, 13.93075736904710736607385637478, 14.95930822858143870174776473777, 15.483888917295256513741041096455, 16.15492910677769897943826123505, 16.89886650296624551926951028520, 17.49977907623962330436955903201, 18.20764740724709167113675315795, 19.13139369089464014756754261685, 20.13690161824267902114948360154, 20.92304237934642467051317988673, 22.04713031958225998849968553591

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