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Properties

Label 1-4235-4235.1003-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.835 + 0.549i$
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  + (0.572 + 0.820i)2-s + (−0.743 + 0.669i)3-s + (−0.345 + 0.938i)4-s + (−0.974 − 0.226i)6-s + (−0.967 + 0.254i)8-s + (0.104 − 0.994i)9-s + (−0.371 − 0.928i)12-s + (−0.717 − 0.696i)13-s + (−0.761 − 0.647i)16-s + (0.508 + 0.861i)17-s + (0.875 − 0.483i)18-s + (−0.217 + 0.976i)19-s + (−0.971 + 0.235i)23-s + (0.548 − 0.836i)24-s + (0.161 − 0.986i)26-s + (0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (0.572 + 0.820i)2-s + (−0.743 + 0.669i)3-s + (−0.345 + 0.938i)4-s + (−0.974 − 0.226i)6-s + (−0.967 + 0.254i)8-s + (0.104 − 0.994i)9-s + (−0.371 − 0.928i)12-s + (−0.717 − 0.696i)13-s + (−0.761 − 0.647i)16-s + (0.508 + 0.861i)17-s + (0.875 − 0.483i)18-s + (−0.217 + 0.976i)19-s + (−0.971 + 0.235i)23-s + (0.548 − 0.836i)24-s + (0.161 − 0.986i)26-s + (0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.835 + 0.549i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.835 + 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8413956084 + 0.2520440512i\)
\(L(\frac12)\) \(\approx\) \(0.8413956084 + 0.2520440512i\)
\(L(1)\) \(\approx\) \(0.7103352150 + 0.5367456541i\)
\(L(1)\) \(\approx\) \(0.7103352150 + 0.5367456541i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.572 + 0.820i)T \)
3 \( 1 + (-0.743 + 0.669i)T \)
13 \( 1 + (-0.717 - 0.696i)T \)
17 \( 1 + (0.508 + 0.861i)T \)
19 \( 1 + (-0.217 + 0.976i)T \)
23 \( 1 + (-0.971 + 0.235i)T \)
29 \( 1 + (0.993 + 0.113i)T \)
31 \( 1 + (-0.532 - 0.846i)T \)
37 \( 1 + (-0.556 + 0.830i)T \)
41 \( 1 + (0.921 - 0.389i)T \)
43 \( 1 + (-0.540 - 0.841i)T \)
47 \( 1 + (-0.875 - 0.483i)T \)
53 \( 1 + (-0.647 - 0.761i)T \)
59 \( 1 + (-0.797 + 0.603i)T \)
61 \( 1 + (0.905 - 0.424i)T \)
67 \( 1 + (-0.189 - 0.981i)T \)
71 \( 1 + (0.198 + 0.980i)T \)
73 \( 1 + (0.475 - 0.879i)T \)
79 \( 1 + (-0.625 + 0.780i)T \)
83 \( 1 + (-0.791 - 0.610i)T \)
89 \( 1 + (-0.580 + 0.814i)T \)
97 \( 1 + (-0.0570 + 0.998i)T \)
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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

−18.308841146831992930965064816870, −17.85259454037956814909897808304, −17.155990148622352543524672512438, −16.13913976388885342747731932970, −15.7884820818539277147202520678, −14.424526589948897638466413622624, −14.24821457781725524304526616867, −13.41483549371973341607414040561, −12.67661529375188028352255699388, −12.20049733425352140245060770491, −11.5559746858586404166956833763, −11.05041901576876113839668203697, −10.222735733639013056657882303645, −9.595433801752429806898477399134, −8.7716535791891093558255626849, −7.74217300625912528664353115078, −6.92267613322968120191798033929, −6.35464741678116888138406140455, −5.519702074485899619591545477399, −4.79505524685646786644509625245, −4.35472802896612880723574503472, −3.09886186515015458549759911022, −2.42151012415947801157983825486, −1.65394953368165090054082277654, −0.76341561284565019994276546517, 0.28934596642681891872790211099, 1.78330547416506153846923540359, 3.04289910336056929474551304991, 3.724661209268947517293577484192, 4.35636787990653892218427356477, 5.19687802429717693853930829771, 5.73009095969597425463482424076, 6.31575437197248095080772038363, 7.127412766476656998939509720111, 8.02888871700937868601730359070, 8.4932845073926596789240905438, 9.68401595089031742705436367537, 10.049166696018469397516564756530, 10.92137221214785802433023070157, 11.85969663003947425772945300297, 12.349210021481351996566098262144, 12.851412195400471312769182714140, 13.89697695075068561714422369799, 14.55898470161708312798723942527, 15.136266493979657056278379191365, 15.67675644935162501740043526672, 16.45840704642032139497201488061, 16.90492488496746697699446710781, 17.53142152190989654662205277354, 18.084252598084296289454669239867

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