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Properties

Label 1-4235-4235.1063-r1-0-0
Degree $1$
Conductor $4235$
Sign $0.816 + 0.576i$
Analytic cond. $455.113$
Root an. cond. $455.113$
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  + (0.441 + 0.897i)2-s + (−0.587 + 0.809i)3-s + (−0.610 + 0.791i)4-s + (−0.985 − 0.170i)6-s + (−0.980 − 0.198i)8-s + (−0.309 − 0.951i)9-s + (−0.281 − 0.959i)12-s + (0.825 − 0.564i)13-s + (−0.254 − 0.967i)16-s + (0.389 + 0.921i)17-s + (0.717 − 0.696i)18-s + (−0.974 + 0.226i)19-s + (0.540 + 0.841i)23-s + (0.736 − 0.676i)24-s + (0.870 + 0.491i)26-s + (0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (0.441 + 0.897i)2-s + (−0.587 + 0.809i)3-s + (−0.610 + 0.791i)4-s + (−0.985 − 0.170i)6-s + (−0.980 − 0.198i)8-s + (−0.309 − 0.951i)9-s + (−0.281 − 0.959i)12-s + (0.825 − 0.564i)13-s + (−0.254 − 0.967i)16-s + (0.389 + 0.921i)17-s + (0.717 − 0.696i)18-s + (−0.974 + 0.226i)19-s + (0.540 + 0.841i)23-s + (0.736 − 0.676i)24-s + (0.870 + 0.491i)26-s + (0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.816 + 0.576i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1063, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ 0.816 + 0.576i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.052894934 + 0.3342474918i\)
\(L(\frac12)\) \(\approx\) \(1.052894934 + 0.3342474918i\)
\(L(1)\) \(\approx\) \(0.6940871453 + 0.6328040019i\)
\(L(1)\) \(\approx\) \(0.6940871453 + 0.6328040019i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.441 + 0.897i)T \)
3 \( 1 + (-0.587 + 0.809i)T \)
13 \( 1 + (0.825 - 0.564i)T \)
17 \( 1 + (0.389 + 0.921i)T \)
19 \( 1 + (-0.974 + 0.226i)T \)
23 \( 1 + (0.540 + 0.841i)T \)
29 \( 1 + (0.0855 - 0.996i)T \)
31 \( 1 + (0.0285 + 0.999i)T \)
37 \( 1 + (-0.336 - 0.941i)T \)
41 \( 1 + (-0.466 + 0.884i)T \)
43 \( 1 + (0.909 - 0.415i)T \)
47 \( 1 + (0.717 + 0.696i)T \)
53 \( 1 + (-0.967 - 0.254i)T \)
59 \( 1 + (-0.466 - 0.884i)T \)
61 \( 1 + (0.897 + 0.441i)T \)
67 \( 1 + (-0.989 + 0.142i)T \)
71 \( 1 + (0.516 + 0.856i)T \)
73 \( 1 + (-0.931 - 0.362i)T \)
79 \( 1 + (0.993 + 0.113i)T \)
83 \( 1 + (0.633 + 0.774i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (-0.676 - 0.736i)T \)
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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

−18.34859341343934483903321340479, −17.68540532819719876621642059508, −16.880776590413443298824910738846, −16.2470961067045080803845878438, −15.28948763656618046771727515282, −14.493878702894158983156313369809, −13.75063503298669107769656892203, −13.36722797591176730684035592236, −12.53328321382482816982001842823, −12.099686445318009621686567099798, −11.318350831662782915406218637814, −10.83995403452856684133242610155, −10.211203059082934586210843526137, −9.11021499094302929002790251080, −8.65132314308659549274351156023, −7.646005113059160506342300262524, −6.67813136332601119908738196491, −6.243437188589792828907588715734, −5.3368650823459406064882088567, −4.728804372792135327170747696855, −3.913370759194718920735422764736, −2.89620614728805786022727716177, −2.22712926270868578859423162044, −1.37377635622709337891662922669, −0.67912045570281096634953380536, 0.21458625171345630631340409826, 1.33327912969077180226414811973, 2.819793169107550197694614273839, 3.65304528487201136381767543482, 4.10247993333917968763245639273, 4.96036086104929941993617401165, 5.726601519997945358846209832054, 6.11256484479960455366040746626, 6.88907405937167510802172412479, 7.877747738895058310139664397044, 8.51799609174946129695524427172, 9.18028082240732984796078171755, 10.007233069200346027439202722186, 10.75852801796712693354200311308, 11.379090559579174703474956197095, 12.38452535277149547317617747043, 12.747260696589412420858743438626, 13.64269287981838580569047046488, 14.41382631006755880712355522702, 15.075338322092649675861720213, 15.57171069339558190353792744706, 16.130211793859090714628107845473, 16.852985388296279293281611498520, 17.48948119535285894729327202233, 17.78660977537360992195895231482

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