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Properties

Label 1-4235-4235.1913-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.352 - 0.935i$
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  + (−0.945 − 0.327i)2-s + (−0.866 − 0.5i)3-s + (0.786 + 0.618i)4-s + (0.654 + 0.755i)6-s + (−0.540 − 0.841i)8-s + (0.5 + 0.866i)9-s + (−0.371 − 0.928i)12-s + (0.989 + 0.142i)13-s + (0.235 + 0.971i)16-s + (0.0950 − 0.995i)17-s + (−0.189 − 0.981i)18-s + (−0.995 + 0.0950i)19-s + (−0.971 + 0.235i)23-s + (0.0475 + 0.998i)24-s + (−0.888 − 0.458i)26-s − i·27-s + ⋯
L(s)  = 1  + (−0.945 − 0.327i)2-s + (−0.866 − 0.5i)3-s + (0.786 + 0.618i)4-s + (0.654 + 0.755i)6-s + (−0.540 − 0.841i)8-s + (0.5 + 0.866i)9-s + (−0.371 − 0.928i)12-s + (0.989 + 0.142i)13-s + (0.235 + 0.971i)16-s + (0.0950 − 0.995i)17-s + (−0.189 − 0.981i)18-s + (−0.995 + 0.0950i)19-s + (−0.971 + 0.235i)23-s + (0.0475 + 0.998i)24-s + (−0.888 − 0.458i)26-s − i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.352 - 0.935i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1913, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.352 - 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3664407591 - 0.5296848065i\)
\(L(\frac12)\) \(\approx\) \(0.3664407591 - 0.5296848065i\)
\(L(1)\) \(\approx\) \(0.5216906390 - 0.1824508174i\)
\(L(1)\) \(\approx\) \(0.5216906390 - 0.1824508174i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.945 - 0.327i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (0.989 + 0.142i)T \)
17 \( 1 + (0.0950 - 0.995i)T \)
19 \( 1 + (-0.995 + 0.0950i)T \)
23 \( 1 + (-0.971 + 0.235i)T \)
29 \( 1 + (0.415 - 0.909i)T \)
31 \( 1 + (0.928 + 0.371i)T \)
37 \( 1 + (0.618 + 0.786i)T \)
41 \( 1 + (0.654 + 0.755i)T \)
43 \( 1 + (-0.540 - 0.841i)T \)
47 \( 1 + (0.189 - 0.981i)T \)
53 \( 1 + (0.971 + 0.235i)T \)
59 \( 1 + (0.327 + 0.945i)T \)
61 \( 1 + (-0.981 - 0.189i)T \)
67 \( 1 + (-0.189 - 0.981i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (-0.690 - 0.723i)T \)
79 \( 1 + (0.0475 - 0.998i)T \)
83 \( 1 + (0.281 + 0.959i)T \)
89 \( 1 + (-0.580 + 0.814i)T \)
97 \( 1 + (-0.540 - 0.841i)T \)
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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

−18.44800867668119318085220263278, −17.63495194910719616381862684943, −17.434325168485771075110037393283, −16.45713864069212668903464830602, −16.13825898601702431910402685836, −15.42518486841436784660778119687, −14.82319254548721798011212595143, −14.08282942891157273221932065970, −12.8916438444574937032535154317, −12.344972242726557083884117151113, −11.39255259227330080460345987817, −10.94901257542068203350317779870, −10.300540189953854982700136145254, −9.806473006367897626777776037851, −8.80031552559318098920580685555, −8.39455894236528016735095158857, −7.47378785268461882952195979604, −6.512765144556377116247569802522, −6.12110540943290766468313438727, −5.52913226253662150664538140795, −4.44785723294695050637026822306, −3.79794604853202683585978704874, −2.64164414510308630967511724083, −1.59833215778292046298507503742, −0.82305924870923613089338074309, 0.38782054072969063936863159132, 1.22691321382684641251065088342, 2.03462088346617186431398517483, 2.81638730740844559126627740995, 3.92106355613302321714658797338, 4.684353571317384051808544968284, 5.862839544484092455303425430475, 6.34628703682238392219508064275, 6.9958949252939942424802711468, 7.8779579269459125212456615580, 8.33996830186546997752704148539, 9.23720318161302123092729166897, 10.10836256444862296951955788927, 10.56334857088838898862966735392, 11.36474428725249350873167126614, 11.89034925404048547951226262779, 12.35495981745594879956571708141, 13.46488201751807419131252371505, 13.65963541552414681232362385153, 15.116116163019227383150902969, 15.70099977201524254521173892186, 16.49377222104877906299920187972, 16.78499626167846064208307475208, 17.70080684835356717723498746659, 18.16148815771128693399527633943

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