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Properties

Label 1-71e2-5041.1004-r0-0-0
Degree $1$
Conductor $5041$
Sign $0.195 + 0.980i$
Analytic cond. $23.4103$
Root an. cond. $23.4103$
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  + (0.146 − 0.989i)2-s + (−0.707 + 0.706i)3-s + (−0.956 − 0.290i)4-s + (0.189 + 0.981i)5-s + (0.595 + 0.803i)6-s + (0.921 − 0.388i)7-s + (−0.427 + 0.903i)8-s + (0.000632 − 0.999i)9-s + (0.999 − 0.0429i)10-s + (0.936 + 0.351i)11-s + (0.882 − 0.471i)12-s + (0.864 + 0.502i)13-s + (−0.249 − 0.968i)14-s + (−0.827 − 0.560i)15-s + (0.831 + 0.555i)16-s + (0.646 + 0.762i)17-s + ⋯
L(s)  = 1  + (0.146 − 0.989i)2-s + (−0.707 + 0.706i)3-s + (−0.956 − 0.290i)4-s + (0.189 + 0.981i)5-s + (0.595 + 0.803i)6-s + (0.921 − 0.388i)7-s + (−0.427 + 0.903i)8-s + (0.000632 − 0.999i)9-s + (0.999 − 0.0429i)10-s + (0.936 + 0.351i)11-s + (0.882 − 0.471i)12-s + (0.864 + 0.502i)13-s + (−0.249 − 0.968i)14-s + (−0.827 − 0.560i)15-s + (0.831 + 0.555i)16-s + (0.646 + 0.762i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5041\)    =    \(71^{2}\)
Sign: $0.195 + 0.980i$
Analytic conductor: \(23.4103\)
Root analytic conductor: \(23.4103\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5041} (1004, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5041,\ (0:\ ),\ 0.195 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.034356343 + 0.8487462163i\)
\(L(\frac12)\) \(\approx\) \(1.034356343 + 0.8487462163i\)
\(L(1)\) \(\approx\) \(0.9785330954 + 0.002809733369i\)
\(L(1)\) \(\approx\) \(0.9785330954 + 0.002809733369i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (0.146 - 0.989i)T \)
3 \( 1 + (-0.707 + 0.706i)T \)
5 \( 1 + (0.189 + 0.981i)T \)
7 \( 1 + (0.921 - 0.388i)T \)
11 \( 1 + (0.936 + 0.351i)T \)
13 \( 1 + (0.864 + 0.502i)T \)
17 \( 1 + (0.646 + 0.762i)T \)
19 \( 1 + (-0.386 - 0.922i)T \)
23 \( 1 + (0.652 - 0.757i)T \)
29 \( 1 + (-0.975 - 0.221i)T \)
31 \( 1 + (-0.855 + 0.518i)T \)
37 \( 1 + (-0.259 + 0.965i)T \)
41 \( 1 + (0.313 + 0.949i)T \)
43 \( 1 + (0.539 + 0.842i)T \)
47 \( 1 + (-0.995 + 0.0997i)T \)
53 \( 1 + (0.0511 + 0.998i)T \)
59 \( 1 + (-0.999 + 0.00379i)T \)
61 \( 1 + (-0.996 + 0.0871i)T \)
67 \( 1 + (-0.182 + 0.983i)T \)
73 \( 1 + (-0.360 + 0.932i)T \)
79 \( 1 + (0.958 - 0.284i)T \)
83 \( 1 + (-0.559 + 0.828i)T \)
89 \( 1 + (0.340 - 0.940i)T \)
97 \( 1 + (-0.0347 - 0.999i)T \)
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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

−17.58991750708986303375634357676, −17.305081298430962047880935580172, −16.44479298125913018695691973141, −16.294628947659556039064506635990, −15.26624072457626216944734601549, −14.5335958718396533348677929985, −13.81971477052151275028096959233, −13.36125721373296978744209495379, −12.48233721241836600400206663862, −12.15771773799640142502846528692, −11.32945537556424060350848801283, −10.61056975167449727703966357615, −9.294294232257363499859103666165, −9.03788583616599072193949464779, −8.043274556306365975537255303486, −7.77208473467235987184054738161, −6.894873477036144656346145337837, −5.917107951915410814006577482800, −5.58761848506446485357876838203, −5.11706463545438139758273092972, −4.16889462958878490934730173486, −3.447373191241030444043176787106, −1.85603409899118325979788667045, −1.312891055547363200978778045458, −0.40911710170558801639008218383, 1.2002443732736519725460604363, 1.61725565797958205059956436654, 2.81537078439810381161999020144, 3.567755398413350898623740288209, 4.238106941398456704128365880537, 4.69469538669424969591007610153, 5.69079225704384673849795376220, 6.32985637277720192548188208961, 7.05557101721318224137791727773, 8.160168828905614746387942668493, 9.024349756304609785511565683423, 9.55141866646059844431514329590, 10.385798661174227914903753247269, 10.96251075733442113221249636382, 11.20592348247709030638442882433, 11.8555581744938191457048737541, 12.67887406192559196022534452892, 13.50556696102987158016744562374, 14.3086541809504154157123859548, 14.82071243199436053047863025912, 15.11503792239083493775266867101, 16.386303519494396668617384720535, 17.17468583760026788390246400371, 17.45838634398416163766422183572, 18.31028073783797702614132559880

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