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Properties

Label 1-2527-2527.137-r0-0-0
Degree $1$
Conductor $2527$
Sign $-0.189 - 0.981i$
Analytic cond. $11.7353$
Root an. cond. $11.7353$
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  + (−0.227 + 0.973i)2-s + (0.515 + 0.856i)3-s + (−0.896 − 0.443i)4-s + (−0.991 + 0.128i)5-s + (−0.951 + 0.307i)6-s + (0.635 − 0.771i)8-s + (−0.467 + 0.883i)9-s + (0.100 − 0.994i)10-s + (0.789 − 0.614i)11-s + (−0.0825 − 0.996i)12-s + (−0.777 − 0.628i)13-s + (−0.621 − 0.783i)15-s + (0.606 + 0.794i)16-s + (0.832 + 0.554i)17-s + (−0.754 − 0.656i)18-s + ⋯
L(s)  = 1  + (−0.227 + 0.973i)2-s + (0.515 + 0.856i)3-s + (−0.896 − 0.443i)4-s + (−0.991 + 0.128i)5-s + (−0.951 + 0.307i)6-s + (0.635 − 0.771i)8-s + (−0.467 + 0.883i)9-s + (0.100 − 0.994i)10-s + (0.789 − 0.614i)11-s + (−0.0825 − 0.996i)12-s + (−0.777 − 0.628i)13-s + (−0.621 − 0.783i)15-s + (0.606 + 0.794i)16-s + (0.832 + 0.554i)17-s + (−0.754 − 0.656i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2527\)    =    \(7 \cdot 19^{2}\)
Sign: $-0.189 - 0.981i$
Analytic conductor: \(11.7353\)
Root analytic conductor: \(11.7353\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2527} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2527,\ (0:\ ),\ -0.189 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2422300509 + 0.2935569316i\)
\(L(\frac12)\) \(\approx\) \(-0.2422300509 + 0.2935569316i\)
\(L(1)\) \(\approx\) \(0.5262744775 + 0.5190975431i\)
\(L(1)\) \(\approx\) \(0.5262744775 + 0.5190975431i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.227 + 0.973i)T \)
3 \( 1 + (0.515 + 0.856i)T \)
5 \( 1 + (-0.991 + 0.128i)T \)
11 \( 1 + (0.789 - 0.614i)T \)
13 \( 1 + (-0.777 - 0.628i)T \)
17 \( 1 + (0.832 + 0.554i)T \)
23 \( 1 + (-0.999 - 0.0183i)T \)
29 \( 1 + (0.280 + 0.959i)T \)
31 \( 1 + (-0.998 + 0.0550i)T \)
37 \( 1 + (0.137 + 0.990i)T \)
41 \( 1 + (0.663 - 0.748i)T \)
43 \( 1 + (-0.912 - 0.410i)T \)
47 \( 1 + (0.741 + 0.670i)T \)
53 \( 1 + (0.209 - 0.977i)T \)
59 \( 1 + (0.315 + 0.948i)T \)
61 \( 1 + (0.983 + 0.182i)T \)
67 \( 1 + (0.870 + 0.492i)T \)
71 \( 1 + (-0.333 + 0.942i)T \)
73 \( 1 + (-0.896 + 0.443i)T \)
79 \( 1 + (-0.912 - 0.410i)T \)
83 \( 1 + (-0.298 + 0.954i)T \)
89 \( 1 + (-0.896 - 0.443i)T \)
97 \( 1 + (-0.861 - 0.507i)T \)
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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

−19.07688763168356896983747556525, −18.4733184556535232502055509483, −17.76426537717587286031748350997, −16.96216647895658500669692753654, −16.29113393499737847230178248156, −15.07770731690241879837282380965, −14.39571544730491266219125825713, −13.946101713102214714438346756352, −12.84111937478924086374049727367, −12.36049865623533639716579403009, −11.75066846272957948983563120580, −11.38712287506756202784632695892, −10.07197667936551783535570558951, −9.41554077869303938916567898192, −8.79655835472393558018179884386, −7.81257308315388814482024857262, −7.52080266843263333711617021571, −6.608901674502606361891258535943, −5.344982276458851020736282726160, −4.24822423413833880414161513851, −3.81380101080004381274278039828, −2.82874310900592805395959270091, −2.052270143418599303587054121348, −1.20674810382805915101863488531, −0.140583136151335529175681454353, 1.25407450917981489744238383217, 2.80665879397175659519550373852, 3.76065741876296516887909534770, 4.090745571465057487324539854360, 5.17509576522515294315813923708, 5.73421044583426491222278954719, 6.89886148294828775187690454673, 7.57673576220127415613265503935, 8.37972152631556799265957757122, 8.68447474673032748937534164320, 9.7501036093895178241121478241, 10.27467806426038290224764637946, 11.09860093662464270539715623220, 12.05673657727407896305953752501, 12.88359410612661332301645804524, 13.94684728002462821003934476181, 14.62977374368215897984727784980, 14.83304902824571768758008252460, 15.77585874343541041449010184912, 16.258906541916282731665183518, 16.840909371471759229523148074138, 17.58390965111299567788985520925, 18.66201246070349909284398166007, 19.21867035798766235616727925147, 19.85206605783622436458371229027

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