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Properties

Label 1-253-253.163-r0-0-0
Degree $1$
Conductor $253$
Sign $0.841 - 0.540i$
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  + (0.198 − 0.980i)2-s + (−0.0285 + 0.999i)3-s + (−0.921 − 0.389i)4-s + (−0.998 + 0.0570i)5-s + (0.974 + 0.226i)6-s + (0.897 − 0.441i)7-s + (−0.564 + 0.825i)8-s + (−0.998 − 0.0570i)9-s + (−0.142 + 0.989i)10-s + (0.415 − 0.909i)12-s + (0.696 − 0.717i)13-s + (−0.254 − 0.967i)14-s + (−0.0285 − 0.999i)15-s + (0.696 + 0.717i)16-s + (0.974 + 0.226i)17-s + (−0.254 + 0.967i)18-s + ⋯
L(s)  = 1  + (0.198 − 0.980i)2-s + (−0.0285 + 0.999i)3-s + (−0.921 − 0.389i)4-s + (−0.998 + 0.0570i)5-s + (0.974 + 0.226i)6-s + (0.897 − 0.441i)7-s + (−0.564 + 0.825i)8-s + (−0.998 − 0.0570i)9-s + (−0.142 + 0.989i)10-s + (0.415 − 0.909i)12-s + (0.696 − 0.717i)13-s + (−0.254 − 0.967i)14-s + (−0.0285 − 0.999i)15-s + (0.696 + 0.717i)16-s + (0.974 + 0.226i)17-s + (−0.254 + 0.967i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $0.841 - 0.540i$
Analytic conductor: \(1.17492\)
Root analytic conductor: \(1.17492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{253} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 253,\ (0:\ ),\ 0.841 - 0.540i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.062490158 - 0.3116498072i\)
\(L(\frac12)\) \(\approx\) \(1.062490158 - 0.3116498072i\)
\(L(1)\) \(\approx\) \(0.9827611470 - 0.2352011888i\)
\(L(1)\) \(\approx\) \(0.9827611470 - 0.2352011888i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.198 - 0.980i)T \)
3 \( 1 + (-0.0285 + 0.999i)T \)
5 \( 1 + (-0.998 + 0.0570i)T \)
7 \( 1 + (0.897 - 0.441i)T \)
13 \( 1 + (0.696 - 0.717i)T \)
17 \( 1 + (0.974 + 0.226i)T \)
19 \( 1 + (0.516 + 0.856i)T \)
29 \( 1 + (0.516 - 0.856i)T \)
31 \( 1 + (0.610 + 0.791i)T \)
37 \( 1 + (0.774 + 0.633i)T \)
41 \( 1 + (0.774 - 0.633i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (-0.466 + 0.884i)T \)
59 \( 1 + (-0.985 + 0.170i)T \)
61 \( 1 + (0.941 - 0.336i)T \)
67 \( 1 + (0.415 + 0.909i)T \)
71 \( 1 + (-0.870 - 0.491i)T \)
73 \( 1 + (-0.921 - 0.389i)T \)
79 \( 1 + (0.696 - 0.717i)T \)
83 \( 1 + (-0.362 + 0.931i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (-0.362 - 0.931i)T \)
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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

−25.872438566676087901929628352150, −24.94514451202704895206457487761, −24.13507187418197753315153192672, −23.60450249541416874387644730391, −22.89932290806210403980327143541, −21.71352212729877841548647203458, −20.53103312111830781434243877091, −19.24851359903834750945273273198, −18.49991162109469606733436201708, −17.80694191065201971290510348455, −16.68038423474562740016324075549, −15.79908475570017701473233151007, −14.70381730452012130627068398243, −14.041093742503208553078801746207, −12.91276737509621492130349172685, −11.93698343331627655607827765780, −11.25313276880281350742413659179, −9.16278233411829954087735782710, −8.24864750045493040779579911115, −7.61926476537113082642474288229, −6.658775386888316618016614463918, −5.49105680575332229506841482296, −4.42512896796050184199263270853, −3.00824934981531131612434387471, −1.085486938796152465037120021926, 1.07466202351102894717573131829, 3.02668034010596618856052665843, 3.85128876899249156979585970432, 4.69885794119394237658332396416, 5.74873789310631815170522042371, 7.91586569851715843062941310313, 8.51362221591460590891288865856, 9.97616957022307347241045126244, 10.65938011127270250569411616974, 11.516739009518331708982405606541, 12.23958222218163431154495275898, 13.754683343576210260315326000767, 14.608381275687937030056379786156, 15.40837530948293317649278323162, 16.561534023296834367689888787623, 17.65212380396045201759577331713, 18.69574302159987114180460228137, 19.77851691443257760793379179929, 20.56273827124876587740278847549, 21.05137366258554635447868646522, 22.14589457563457259187220359642, 23.281371603801076380165579552028, 23.33931248753184675791122528761, 24.98916586539728329093360540379, 26.53196925024481015700774361640

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