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Properties

Label 1-5225-5225.1356-r0-0-0
Degree $1$
Conductor $5225$
Sign $0.172 + 0.984i$
Analytic cond. $24.2647$
Root an. cond. $24.2647$
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  + (−0.978 − 0.207i)2-s + (−0.978 − 0.207i)3-s + (0.913 + 0.406i)4-s + (0.913 + 0.406i)6-s + (−0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (0.913 + 0.406i)9-s + (−0.809 − 0.587i)12-s + (0.913 + 0.406i)13-s + (0.669 + 0.743i)14-s + (0.669 + 0.743i)16-s + (−0.104 − 0.994i)17-s + (−0.809 − 0.587i)18-s + (0.669 + 0.743i)21-s + (0.913 + 0.406i)23-s + (0.669 + 0.743i)24-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (−0.978 − 0.207i)3-s + (0.913 + 0.406i)4-s + (0.913 + 0.406i)6-s + (−0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (0.913 + 0.406i)9-s + (−0.809 − 0.587i)12-s + (0.913 + 0.406i)13-s + (0.669 + 0.743i)14-s + (0.669 + 0.743i)16-s + (−0.104 − 0.994i)17-s + (−0.809 − 0.587i)18-s + (0.669 + 0.743i)21-s + (0.913 + 0.406i)23-s + (0.669 + 0.743i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.172 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.172 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5225\)    =    \(5^{2} \cdot 11 \cdot 19\)
Sign: $0.172 + 0.984i$
Analytic conductor: \(24.2647\)
Root analytic conductor: \(24.2647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5225} (1356, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5225,\ (0:\ ),\ 0.172 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3728593138 + 0.3131005231i\)
\(L(\frac12)\) \(\approx\) \(0.3728593138 + 0.3131005231i\)
\(L(1)\) \(\approx\) \(0.5013392476 - 0.03466180414i\)
\(L(1)\) \(\approx\) \(0.5013392476 - 0.03466180414i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.978 - 0.207i)T \)
3 \( 1 + (-0.978 - 0.207i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (0.913 + 0.406i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.913 + 0.406i)T \)
29 \( 1 + (0.913 + 0.406i)T \)
31 \( 1 + T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.669 - 0.743i)T \)
53 \( 1 + (-0.104 + 0.994i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 + (-0.104 + 0.994i)T \)
67 \( 1 + (0.669 - 0.743i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.913 - 0.406i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (0.913 + 0.406i)T \)
97 \( 1 + (0.913 - 0.406i)T \)
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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

−17.62400906208908096323816451302, −17.32066398571109494753674781307, −16.505066670931674162386245789150, −15.97873720810603066398398201995, −15.40671933063164274109887055237, −15.02104699283343981096065840467, −13.814872141445987460952083615577, −12.87146016827116803342944160223, −12.35698462632122694563617178046, −11.68752566090830593230674752108, −10.87999360890353427958083282929, −10.445526715246591540629409036997, −9.8406568676239985770040926432, −8.994529317763806377187707270599, −8.5211955839868038060963824185, −7.60528438623812101928462475795, −6.673350465685068030393624645595, −6.2831994063460986672249660765, −5.73417079208547853310119623511, −4.933818396316297673219898822544, −3.83082070191243736460892729662, −3.03549031112606018022423030415, −2.058464626257454858030799886284, −1.10962025962918450363013325197, −0.276187208842008842620636363894, 1.02371889117637824485142321460, 1.22963141663634067848309159400, 2.60486684051145582747620423515, 3.267385585916891276787295649449, 4.239174408670194429092062397328, 5.05809121323043986610610753601, 6.17406533780640076294278041823, 6.55019666982917302000265774596, 7.12692323630199817690072012616, 7.854114631398673684468343216944, 8.75060056440073988449757252122, 9.50115507133831699229362779865, 10.1069895688197154381702218699, 10.67780766479487537099592170653, 11.413789641888083306039458389076, 11.81698784331943764587028979726, 12.6410514949911312192713096843, 13.33984707123720407530040150871, 13.85751943128267215100138478370, 15.250488498261903579348408731807, 15.75175250296337005655470585391, 16.4174198605623212980801704933, 16.775401943647034067251246725030, 17.46290327107127386501312877904, 18.13985789135455842161986148722

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