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Properties

Label 1-35e2-1225.137-r1-0-0
Degree $1$
Conductor $1225$
Sign $-0.236 - 0.971i$
Analytic cond. $131.644$
Root an. cond. $131.644$
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  + (−0.999 + 0.0149i)2-s + (0.967 − 0.251i)3-s + (0.999 − 0.0299i)4-s + (−0.963 + 0.266i)6-s + (−0.998 + 0.0448i)8-s + (0.873 − 0.486i)9-s + (−0.873 − 0.486i)11-s + (0.959 − 0.280i)12-s + (−0.0896 + 0.995i)13-s + (0.998 − 0.0598i)16-s + (0.850 + 0.525i)17-s + (−0.866 + 0.5i)18-s + (−0.913 + 0.406i)19-s + (0.880 + 0.473i)22-s + (−0.237 − 0.971i)23-s + (−0.955 + 0.294i)24-s + ⋯
L(s)  = 1  + (−0.999 + 0.0149i)2-s + (0.967 − 0.251i)3-s + (0.999 − 0.0299i)4-s + (−0.963 + 0.266i)6-s + (−0.998 + 0.0448i)8-s + (0.873 − 0.486i)9-s + (−0.873 − 0.486i)11-s + (0.959 − 0.280i)12-s + (−0.0896 + 0.995i)13-s + (0.998 − 0.0598i)16-s + (0.850 + 0.525i)17-s + (−0.866 + 0.5i)18-s + (−0.913 + 0.406i)19-s + (0.880 + 0.473i)22-s + (−0.237 − 0.971i)23-s + (−0.955 + 0.294i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.236 - 0.971i$
Analytic conductor: \(131.644\)
Root analytic conductor: \(131.644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (1:\ ),\ -0.236 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7719671767 - 0.9825786753i\)
\(L(\frac12)\) \(\approx\) \(0.7719671767 - 0.9825786753i\)
\(L(1)\) \(\approx\) \(0.8762716137 - 0.1393405888i\)
\(L(1)\) \(\approx\) \(0.8762716137 - 0.1393405888i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.999 + 0.0149i)T \)
3 \( 1 + (0.967 - 0.251i)T \)
11 \( 1 + (-0.873 - 0.486i)T \)
13 \( 1 + (-0.0896 + 0.995i)T \)
17 \( 1 + (0.850 + 0.525i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (-0.237 - 0.971i)T \)
29 \( 1 + (-0.983 + 0.178i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (0.959 - 0.280i)T \)
41 \( 1 + (0.936 - 0.351i)T \)
43 \( 1 + (-0.781 - 0.623i)T \)
47 \( 1 + (-0.323 - 0.946i)T \)
53 \( 1 + (0.0299 + 0.999i)T \)
59 \( 1 + (0.447 - 0.894i)T \)
61 \( 1 + (0.791 + 0.611i)T \)
67 \( 1 + (-0.994 - 0.104i)T \)
71 \( 1 + (0.473 - 0.880i)T \)
73 \( 1 + (0.817 - 0.575i)T \)
79 \( 1 + (0.104 + 0.994i)T \)
83 \( 1 + (-0.657 - 0.753i)T \)
89 \( 1 + (-0.420 + 0.907i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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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

−20.947250934624696934984171196520, −20.27243961772671271058796333691, −19.66720721141944275064920898018, −18.869820289070116596232291692131, −18.22888675117989552120126825561, −17.47175000729693963062982737248, −16.50120919130812674675101007060, −15.76881728797214519544918328095, −15.079649013889101013674800456678, −14.59047917183307805124570339638, −13.179046673992209206831499611763, −12.81785475162067427309170000749, −11.54211125146401673598746374163, −10.719586473234841529002866484714, −9.801492735024247261223290310526, −9.56320610664723412331695203255, −8.36732116872854270831577654724, −7.78352019216323304497612081624, −7.30060365804934432003905486307, −6.00111214873767692564157393072, −5.014644478855601758135120870710, −3.73682115860011754048037134843, −2.79164269863357755132119393140, −2.17994617405115620128877505196, −0.97928879566471012165048844827, 0.31688721521377558907898897509, 1.58078905216876459161120894262, 2.27563223527806819730733139346, 3.21213599958054636040686771388, 4.16783287799665474266480706536, 5.64187927964293949287036653252, 6.58934410920120575077059514597, 7.3678331268568044957072767630, 8.21682449808402488189420606700, 8.66181984658162328012661721367, 9.57106628080917795028056624168, 10.33235284034254026367926541998, 11.05167633364121853050315742828, 12.24527478807535288614334903540, 12.77984807659996304243788121055, 13.91317545875344140019188843315, 14.67557520404050555866696947068, 15.28596200241694724037179312065, 16.39241368532235125085891617413, 16.67262773055850894744291138255, 17.96092250071594432193489506157, 18.60056649342911383800766186240, 19.078221842059040128764621028673, 19.74489745720082741582765534309, 20.67483910308475921708232744140

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