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Properties

Label 1-4235-4235.689-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.0713 - 0.997i$
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  + (−0.217 − 0.976i)2-s + (−0.104 + 0.994i)3-s + (−0.905 + 0.424i)4-s + (0.993 − 0.113i)6-s + (0.610 + 0.791i)8-s + (−0.978 − 0.207i)9-s + (−0.327 − 0.945i)12-s + (0.921 + 0.389i)13-s + (0.640 − 0.768i)16-s + (0.710 − 0.703i)17-s + (0.00951 + 0.999i)18-s + (0.988 + 0.151i)19-s + (−0.928 − 0.371i)23-s + (−0.851 + 0.524i)24-s + (0.179 − 0.983i)26-s + (0.309 − 0.951i)27-s + ⋯
L(s)  = 1  + (−0.217 − 0.976i)2-s + (−0.104 + 0.994i)3-s + (−0.905 + 0.424i)4-s + (0.993 − 0.113i)6-s + (0.610 + 0.791i)8-s + (−0.978 − 0.207i)9-s + (−0.327 − 0.945i)12-s + (0.921 + 0.389i)13-s + (0.640 − 0.768i)16-s + (0.710 − 0.703i)17-s + (0.00951 + 0.999i)18-s + (0.988 + 0.151i)19-s + (−0.928 − 0.371i)23-s + (−0.851 + 0.524i)24-s + (0.179 − 0.983i)26-s + (0.309 − 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0713 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0713 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.0713 - 0.997i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.0713 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8734552557 - 0.8132469529i\)
\(L(\frac12)\) \(\approx\) \(0.8734552557 - 0.8132469529i\)
\(L(1)\) \(\approx\) \(0.8553967892 - 0.2072899305i\)
\(L(1)\) \(\approx\) \(0.8553967892 - 0.2072899305i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.217 - 0.976i)T \)
3 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (0.921 + 0.389i)T \)
17 \( 1 + (0.710 - 0.703i)T \)
19 \( 1 + (0.988 + 0.151i)T \)
23 \( 1 + (-0.928 - 0.371i)T \)
29 \( 1 + (0.998 - 0.0570i)T \)
31 \( 1 + (0.999 + 0.0190i)T \)
37 \( 1 + (0.683 - 0.730i)T \)
41 \( 1 + (0.198 - 0.980i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (0.00951 - 0.999i)T \)
53 \( 1 + (-0.640 - 0.768i)T \)
59 \( 1 + (-0.749 + 0.662i)T \)
61 \( 1 + (0.953 - 0.299i)T \)
67 \( 1 + (-0.580 + 0.814i)T \)
71 \( 1 + (0.774 - 0.633i)T \)
73 \( 1 + (-0.272 + 0.962i)T \)
79 \( 1 + (-0.997 + 0.0760i)T \)
83 \( 1 + (-0.897 - 0.441i)T \)
89 \( 1 + (-0.0475 - 0.998i)T \)
97 \( 1 + (-0.0285 - 0.999i)T \)
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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

−18.48781702770361460345977029096, −17.72340544921217043154781432127, −17.396989210623782510335120228706, −16.47785233408327002472645704154, −15.95800334049046250409038486159, −15.192721551112279503472650323223, −14.392245095934474530504212667507, −13.7879225287693995523589827930, −13.33132743784317162376080154381, −12.55211944038470454164673051426, −11.854207151298906281816437340398, −11.01209031020455983314186733767, −10.124002776303943352385716862272, −9.47171512926648693963657815266, −8.44230734660496932877613300389, −8.03094693809519954420484244514, −7.56282094615566844403742103781, −6.409113832499453548619658077, −6.262212749594590335630155507412, −5.437392826175752636035515131725, −4.64111109198522229863112268856, −3.57082685900102216623491763866, −2.78988702429541656622172362366, −1.348361414712345133303611869933, −1.07643610265156035269068586024, 0.44227004318448151546316252455, 1.420294271177467930313304203477, 2.53219567368744096170422858468, 3.21152320647730501658994107472, 3.8990106238943827950712829953, 4.541564984943250862336729388859, 5.34306233034369628860576920746, 6.00962566801481131298194619182, 7.17328543917938625545601959035, 8.24637770701060008899455067961, 8.61215872449074256138284382381, 9.541647650432454226132150187577, 9.98433080045438163915783246898, 10.55067021202449224243019664093, 11.48947116965772338621580935324, 11.74197737345298754886073605103, 12.51054236584035226724087440491, 13.63742972992893986018324348118, 13.994890318060437811691102710471, 14.63848269936756723094899174458, 15.819390951040328673613375151607, 16.11826407607437271453446695126, 16.883169817575881731460434600685, 17.64668996263010786480876695934, 18.28167532880682689382849674310

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