Properties

Label 1-4235-4235.492-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.977 - 0.212i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (0.938 − 0.345i)2-s + (−0.994 + 0.104i)3-s + (0.761 − 0.647i)4-s + (−0.897 + 0.441i)6-s + (0.491 − 0.870i)8-s + (0.978 − 0.207i)9-s + (−0.690 + 0.723i)12-s + (−0.999 − 0.0285i)13-s + (0.161 − 0.986i)16-s + (0.875 − 0.483i)17-s + (0.846 − 0.532i)18-s + (−0.905 + 0.424i)19-s + (−0.458 + 0.888i)23-s + (−0.398 + 0.917i)24-s + (−0.948 + 0.318i)26-s + (−0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (0.938 − 0.345i)2-s + (−0.994 + 0.104i)3-s + (0.761 − 0.647i)4-s + (−0.897 + 0.441i)6-s + (0.491 − 0.870i)8-s + (0.978 − 0.207i)9-s + (−0.690 + 0.723i)12-s + (−0.999 − 0.0285i)13-s + (0.161 − 0.986i)16-s + (0.875 − 0.483i)17-s + (0.846 − 0.532i)18-s + (−0.905 + 0.424i)19-s + (−0.458 + 0.888i)23-s + (−0.398 + 0.917i)24-s + (−0.948 + 0.318i)26-s + (−0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.977 - 0.212i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (492, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.977 - 0.212i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1145032631 - 1.063462226i\)
\(L(\frac12)\) \(\approx\) \(0.1145032631 - 1.063462226i\)
\(L(1)\) \(\approx\) \(1.097806627 - 0.4135433075i\)
\(L(1)\) \(\approx\) \(1.097806627 - 0.4135433075i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.938 - 0.345i)T \)
3 \( 1 + (-0.994 + 0.104i)T \)
13 \( 1 + (-0.999 - 0.0285i)T \)
17 \( 1 + (0.875 - 0.483i)T \)
19 \( 1 + (-0.905 + 0.424i)T \)
23 \( 1 + (-0.458 + 0.888i)T \)
29 \( 1 + (0.974 - 0.226i)T \)
31 \( 1 + (-0.432 - 0.901i)T \)
37 \( 1 + (0.924 + 0.380i)T \)
41 \( 1 + (-0.696 - 0.717i)T \)
43 \( 1 + (-0.909 + 0.415i)T \)
47 \( 1 + (-0.846 - 0.532i)T \)
53 \( 1 + (0.986 - 0.161i)T \)
59 \( 1 + (-0.272 - 0.962i)T \)
61 \( 1 + (-0.640 - 0.768i)T \)
67 \( 1 + (-0.371 + 0.928i)T \)
71 \( 1 + (-0.921 - 0.389i)T \)
73 \( 1 + (0.836 + 0.548i)T \)
79 \( 1 + (-0.217 + 0.976i)T \)
83 \( 1 + (-0.967 - 0.254i)T \)
89 \( 1 + (0.327 - 0.945i)T \)
97 \( 1 + (0.113 + 0.993i)T \)
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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

−18.50831976770746928570917943782, −17.82637512478261592795296545701, −17.11224536097765796210615449786, −16.55694843934504782788707163864, −16.18027926535636270210947304273, −15.103230362524858720842872333333, −14.79939388691965195044750518348, −13.93439117459535234846711938531, −13.0975478110338819551559217799, −12.503930780145383581900630802494, −12.068817009114482871308438500528, −11.42025612595617982304106551440, −10.42010737910134568767046744988, −10.23977872592376592573775120885, −8.90402422724167915846622391039, −7.98933032586781761308096539422, −7.34497944450251552597105703186, −6.55102985981635522666460310943, −6.14640744876560161024916930607, −5.21723379754383661450590056441, −4.71992268813455432479157758146, −4.06669990507511433019926709290, −3.0389161230870122594974281098, −2.19065307552855300108261768063, −1.2582199182058600613698737048, 0.232237360820816560708563533729, 1.37274997948552382383620508777, 2.14676790428060877534061694348, 3.13832867994098613939724750970, 3.98393204133907434555744884449, 4.66964184361445945212025352908, 5.32244618430805380448990652750, 5.91942056437154712577588611646, 6.665854506717060670638958562050, 7.33282026894029210159781231913, 8.12050071820887071562866868952, 9.60392424547587503446483889396, 9.954417652490290406925084131347, 10.59084606869801334130932781286, 11.591395198424483097920609141079, 11.77888187977616024268097202146, 12.56259516431010953941619481415, 13.11106409714018119501483208104, 13.93088421995624086381089799582, 14.72155379010614879420881291574, 15.25407544967417784297154379999, 15.99034795672681270901494021102, 16.737992417363754374834264050049, 17.10873427409337120497190166588, 18.145709642283602472613616884458

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