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Properties

Label 1-4235-4235.1112-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.631 - 0.775i$
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  + (0.755 − 0.654i)2-s + i·3-s + (0.142 − 0.989i)4-s + (0.654 + 0.755i)6-s + (−0.540 − 0.841i)8-s − 9-s + (0.989 + 0.142i)12-s + (−0.989 − 0.142i)13-s + (−0.959 − 0.281i)16-s + (0.909 − 0.415i)17-s + (−0.755 + 0.654i)18-s + (0.415 − 0.909i)19-s + (−0.281 + 0.959i)23-s + (0.841 − 0.540i)24-s + (−0.841 + 0.540i)26-s − i·27-s + ⋯
L(s)  = 1  + (0.755 − 0.654i)2-s + i·3-s + (0.142 − 0.989i)4-s + (0.654 + 0.755i)6-s + (−0.540 − 0.841i)8-s − 9-s + (0.989 + 0.142i)12-s + (−0.989 − 0.142i)13-s + (−0.959 − 0.281i)16-s + (0.909 − 0.415i)17-s + (−0.755 + 0.654i)18-s + (0.415 − 0.909i)19-s + (−0.281 + 0.959i)23-s + (0.841 − 0.540i)24-s + (−0.841 + 0.540i)26-s − i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.631 - 0.775i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.631 - 0.775i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.044853781 - 0.9719414731i\)
\(L(\frac12)\) \(\approx\) \(2.044853781 - 0.9719414731i\)
\(L(1)\) \(\approx\) \(1.421415251 - 0.2863002340i\)
\(L(1)\) \(\approx\) \(1.421415251 - 0.2863002340i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.755 - 0.654i)T \)
3 \( 1 + iT \)
13 \( 1 + (-0.989 - 0.142i)T \)
17 \( 1 + (0.909 - 0.415i)T \)
19 \( 1 + (0.415 - 0.909i)T \)
23 \( 1 + (-0.281 + 0.959i)T \)
29 \( 1 + (-0.415 + 0.909i)T \)
31 \( 1 + (0.142 + 0.989i)T \)
37 \( 1 + (0.989 - 0.142i)T \)
41 \( 1 + (0.654 + 0.755i)T \)
43 \( 1 + (-0.540 - 0.841i)T \)
47 \( 1 + (0.755 + 0.654i)T \)
53 \( 1 + (0.281 + 0.959i)T \)
59 \( 1 + (-0.654 + 0.755i)T \)
61 \( 1 + (0.654 - 0.755i)T \)
67 \( 1 + (0.755 - 0.654i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (0.281 - 0.959i)T \)
79 \( 1 + (-0.841 - 0.540i)T \)
83 \( 1 + (-0.281 - 0.959i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
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

−18.534928190029103571266142558387, −17.627466494937977860440538530711, −16.926010405291473770691439703606, −16.6741919194162433732611310725, −15.71746182586890064212198113617, −14.63895123848922221812149407318, −14.56922360681600024735814178730, −13.79269492315081773030661579188, −12.92567848425452260538392668893, −12.592726570391567353687104032351, −11.81353014880468783543083112696, −11.41973636020876824591427851009, −10.18728499770840315185699973039, −9.40900222383133012238268893070, −8.28388309039683791132124226284, −7.95134676097444926445019275657, −7.279931428457105648444112820652, −6.56883890547467751942016862567, −5.820239082298083127255159237430, −5.379748083384669365870137993013, −4.31172607216721270386198807451, −3.598533250973584095836536804449, −2.563844895291025452875764060183, −2.12563608530310884925310976712, −0.81880444564680727543295189358, 0.59546376413525326466944940669, 1.74299809534933693542585534741, 2.88173965308590256611644080419, 3.1238691281750943697214183138, 4.098691205287576277134421261120, 4.84195666856259742273353658333, 5.31268114113305845298482556060, 5.986435739893049909312493295939, 7.05185925729759181740740817122, 7.793400214591501850235029127890, 9.06609862411871405749466933109, 9.45061055253304893714356757947, 10.12038556210450707471822695172, 10.78052055269094599505259269202, 11.44602683992100667053674732799, 12.0726685468121405296836219840, 12.702441679086811861207346049274, 13.723978500753882074917725785576, 14.16024199823220539359066593252, 14.87351701893520092550272740934, 15.40442926689817630785196718732, 16.08699388561590979184941675309, 16.75483062905144291166607865068, 17.612974542754067544680793970619, 18.33155780685410703141347570895

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