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Properties

Label 1-253-253.182-r0-0-0
Degree $1$
Conductor $253$
Sign $-0.0706 + 0.997i$
Analytic cond. $1.17492$
Root an. cond. $1.17492$
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  + (0.870 + 0.491i)2-s + (−0.564 − 0.825i)3-s + (0.516 + 0.856i)4-s + (0.362 + 0.931i)5-s + (−0.0855 − 0.996i)6-s + (−0.985 − 0.170i)7-s + (0.0285 + 0.999i)8-s + (−0.362 + 0.931i)9-s + (−0.142 + 0.989i)10-s + (0.415 − 0.909i)12-s + (0.466 + 0.884i)13-s + (−0.774 − 0.633i)14-s + (0.564 − 0.825i)15-s + (−0.466 + 0.884i)16-s + (0.0855 + 0.996i)17-s + (−0.774 + 0.633i)18-s + ⋯
L(s)  = 1  + (0.870 + 0.491i)2-s + (−0.564 − 0.825i)3-s + (0.516 + 0.856i)4-s + (0.362 + 0.931i)5-s + (−0.0855 − 0.996i)6-s + (−0.985 − 0.170i)7-s + (0.0285 + 0.999i)8-s + (−0.362 + 0.931i)9-s + (−0.142 + 0.989i)10-s + (0.415 − 0.909i)12-s + (0.466 + 0.884i)13-s + (−0.774 − 0.633i)14-s + (0.564 − 0.825i)15-s + (−0.466 + 0.884i)16-s + (0.0855 + 0.996i)17-s + (−0.774 + 0.633i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0706 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0706 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $-0.0706 + 0.997i$
Analytic conductor: \(1.17492\)
Root analytic conductor: \(1.17492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{253} (182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 253,\ (0:\ ),\ -0.0706 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9939225019 + 1.066779951i\)
\(L(\frac12)\) \(\approx\) \(0.9939225019 + 1.066779951i\)
\(L(1)\) \(\approx\) \(1.209879006 + 0.5361712608i\)
\(L(1)\) \(\approx\) \(1.209879006 + 0.5361712608i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.870 + 0.491i)T \)
3 \( 1 + (-0.564 - 0.825i)T \)
5 \( 1 + (0.362 + 0.931i)T \)
7 \( 1 + (-0.985 - 0.170i)T \)
13 \( 1 + (0.466 + 0.884i)T \)
17 \( 1 + (0.0855 + 0.996i)T \)
19 \( 1 + (-0.921 - 0.389i)T \)
29 \( 1 + (0.921 - 0.389i)T \)
31 \( 1 + (0.941 - 0.336i)T \)
37 \( 1 + (0.254 + 0.967i)T \)
41 \( 1 + (0.254 - 0.967i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.696 - 0.717i)T \)
59 \( 1 + (0.897 + 0.441i)T \)
61 \( 1 + (0.610 + 0.791i)T \)
67 \( 1 + (-0.415 - 0.909i)T \)
71 \( 1 + (0.198 - 0.980i)T \)
73 \( 1 + (-0.516 - 0.856i)T \)
79 \( 1 + (-0.466 - 0.884i)T \)
83 \( 1 + (-0.998 - 0.0570i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + (0.998 - 0.0570i)T \)
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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

−25.38407574775988623296284693428, −24.92019495405916773959329017966, −23.379258513372197129377679072939, −23.0596507302115314862249630118, −21.981433997040744536110497096733, −21.293900719030802553751081070216, −20.44329910121598681634103888370, −19.75343384932081827442003300362, −18.389127621646396237461122686100, −17.13322548365313620110009438703, −16.03830032268651305818554219343, −15.71423219035535636546573819281, −14.40398903139419107276187406554, −13.20325548497035445565249405013, −12.52187493815672853948716268034, −11.62637988683204896478239066129, −10.359541740733364665631427103638, −9.78123710788990933986178756980, −8.67092272259969672610998890969, −6.591517054820916737739855024590, −5.73917074036551961454996373402, −4.91663079376359819285252093692, −3.84972495129443649033213237356, −2.72894730448210457026703942633, −0.82410819818868308893590772090, 2.00512952740288485365269308959, 3.1185784364174613968735461338, 4.43820231037213787262799861971, 6.14941602828408900363805960775, 6.3202007833951404592793558357, 7.24315496637620115615912975977, 8.49636342613545725553045951099, 10.23932283530466005178282130396, 11.2191352832546855957356562908, 12.19420612056908515743903262226, 13.22723352783229743434899543605, 13.74480828591262742965269161574, 14.84590080085053847944254779518, 15.9406879163300474525683410274, 16.94470339341607643198102576472, 17.63101668408288283680042943359, 18.918018164475072118244668997190, 19.49633997346248308062359626830, 21.13418796541913128190372131548, 21.99051198742459952095239541636, 22.697596215841125246307340315735, 23.45099695557061223089950491794, 24.11527733706495389057669889875, 25.52611294622334174144577355114, 25.69255289110213704175218929724

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