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Properties

Label 1-4235-4235.738-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.964 + 0.263i$
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  + (0.814 − 0.580i)2-s + (−0.866 + 0.5i)3-s + (0.327 − 0.945i)4-s + (−0.415 + 0.909i)6-s + (−0.281 − 0.959i)8-s + (0.5 − 0.866i)9-s + (0.189 + 0.981i)12-s + (−0.755 + 0.654i)13-s + (−0.786 − 0.618i)16-s + (0.998 + 0.0475i)17-s + (−0.0950 − 0.995i)18-s + (0.0475 + 0.998i)19-s + (0.618 − 0.786i)23-s + (0.723 + 0.690i)24-s + (−0.235 + 0.971i)26-s + i·27-s + ⋯
L(s)  = 1  + (0.814 − 0.580i)2-s + (−0.866 + 0.5i)3-s + (0.327 − 0.945i)4-s + (−0.415 + 0.909i)6-s + (−0.281 − 0.959i)8-s + (0.5 − 0.866i)9-s + (0.189 + 0.981i)12-s + (−0.755 + 0.654i)13-s + (−0.786 − 0.618i)16-s + (0.998 + 0.0475i)17-s + (−0.0950 − 0.995i)18-s + (0.0475 + 0.998i)19-s + (0.618 − 0.786i)23-s + (0.723 + 0.690i)24-s + (−0.235 + 0.971i)26-s + i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.964 + 0.263i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (738, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.964 + 0.263i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05688371625 - 0.4240185468i\)
\(L(\frac12)\) \(\approx\) \(-0.05688371625 - 0.4240185468i\)
\(L(1)\) \(\approx\) \(0.9821476951 - 0.3105686317i\)
\(L(1)\) \(\approx\) \(0.9821476951 - 0.3105686317i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.814 - 0.580i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-0.755 + 0.654i)T \)
17 \( 1 + (0.998 + 0.0475i)T \)
19 \( 1 + (0.0475 + 0.998i)T \)
23 \( 1 + (0.618 - 0.786i)T \)
29 \( 1 + (-0.841 + 0.540i)T \)
31 \( 1 + (-0.981 - 0.189i)T \)
37 \( 1 + (-0.945 + 0.327i)T \)
41 \( 1 + (-0.415 + 0.909i)T \)
43 \( 1 + (-0.281 - 0.959i)T \)
47 \( 1 + (0.0950 - 0.995i)T \)
53 \( 1 + (-0.618 - 0.786i)T \)
59 \( 1 + (0.580 - 0.814i)T \)
61 \( 1 + (0.995 + 0.0950i)T \)
67 \( 1 + (0.0950 + 0.995i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (-0.371 - 0.928i)T \)
79 \( 1 + (-0.723 + 0.690i)T \)
83 \( 1 + (-0.989 + 0.142i)T \)
89 \( 1 + (-0.888 + 0.458i)T \)
97 \( 1 + (-0.281 - 0.959i)T \)
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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

−18.60421121152930060335819271802, −17.764299424445917735024138094446, −17.22140922172030336626620751413, −16.8630300136954423323000691332, −15.92310419585510939449228673748, −15.48330856406066078221745577628, −14.63105920125221990201824466461, −13.9903553621725103024319817719, −13.09825051862985233179063969497, −12.788766922600712552767922812759, −12.05933931649649653429200952709, −11.40915862471867192256726739907, −10.820705611971985452795905510316, −9.87662637321153333625407230072, −8.96646427387744405553927119794, −7.920235733477625110671405382467, −7.37623903757113411186102654134, −6.94795093123464914912481643497, −5.966132322399638943003450198691, −5.35984910261041006749184356301, −4.98601751260333792302323091149, −3.97916868836416045324692424357, −3.093682983333859631612134052639, −2.28215446962305141113964174795, −1.23896149747390606514494843781, 0.09851267491935134794418536948, 1.3231386440953195129601189266, 2.02763137227352257585055933618, 3.243501842909934206873451292328, 3.76600117061592797038771854706, 4.56792849999226315299983735043, 5.35469470612541241335231454395, 5.642860182746516493778143175259, 6.779471133457615420498905700473, 7.081792489936430172142557097325, 8.41096711368099606681548955750, 9.461745709000307421768718801100, 9.93815873892633122544507013030, 10.54220880947241053245227795646, 11.29717940491848510640637011853, 11.86751490726872519777446813008, 12.48164568019687665001596269976, 12.95235562754365482130204741053, 14.049026930055474952901328404046, 14.671525847905813514900024361856, 15.04860010799860885648897246420, 16.050012383859261748143339662517, 16.62579689650762731006248722020, 17.050955403732944028988702899986, 18.23051165258026665255025482296

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