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Properties

Label 1-4235-4235.1124-r1-0-0
Degree $1$
Conductor $4235$
Sign $0.846 - 0.532i$
Analytic cond. $455.113$
Root an. cond. $455.113$
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  + (−0.905 − 0.424i)2-s + (0.978 − 0.207i)3-s + (0.640 + 0.768i)4-s + (−0.974 − 0.226i)6-s + (−0.254 − 0.967i)8-s + (0.913 − 0.406i)9-s + (0.786 + 0.618i)12-s + (0.696 − 0.717i)13-s + (−0.179 + 0.983i)16-s + (0.00951 + 0.999i)17-s + (−0.999 − 0.0190i)18-s + (−0.953 + 0.299i)19-s + (−0.723 + 0.690i)23-s + (−0.449 − 0.893i)24-s + (−0.935 + 0.353i)26-s + (0.809 − 0.587i)27-s + ⋯
L(s)  = 1  + (−0.905 − 0.424i)2-s + (0.978 − 0.207i)3-s + (0.640 + 0.768i)4-s + (−0.974 − 0.226i)6-s + (−0.254 − 0.967i)8-s + (0.913 − 0.406i)9-s + (0.786 + 0.618i)12-s + (0.696 − 0.717i)13-s + (−0.179 + 0.983i)16-s + (0.00951 + 0.999i)17-s + (−0.999 − 0.0190i)18-s + (−0.953 + 0.299i)19-s + (−0.723 + 0.690i)23-s + (−0.449 − 0.893i)24-s + (−0.935 + 0.353i)26-s + (0.809 − 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.846 - 0.532i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ 0.846 - 0.532i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.266310936 - 0.6539895318i\)
\(L(\frac12)\) \(\approx\) \(2.266310936 - 0.6539895318i\)
\(L(1)\) \(\approx\) \(1.043933882 - 0.2330139303i\)
\(L(1)\) \(\approx\) \(1.043933882 - 0.2330139303i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.905 - 0.424i)T \)
3 \( 1 + (0.978 - 0.207i)T \)
13 \( 1 + (0.696 - 0.717i)T \)
17 \( 1 + (0.00951 + 0.999i)T \)
19 \( 1 + (-0.953 + 0.299i)T \)
23 \( 1 + (-0.723 + 0.690i)T \)
29 \( 1 + (-0.993 - 0.113i)T \)
31 \( 1 + (0.999 - 0.0380i)T \)
37 \( 1 + (0.0665 - 0.997i)T \)
41 \( 1 + (0.921 - 0.389i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (0.999 - 0.0190i)T \)
53 \( 1 + (0.179 + 0.983i)T \)
59 \( 1 + (0.123 + 0.992i)T \)
61 \( 1 + (-0.820 - 0.572i)T \)
67 \( 1 + (0.327 - 0.945i)T \)
71 \( 1 + (0.198 + 0.980i)T \)
73 \( 1 + (-0.851 + 0.524i)T \)
79 \( 1 + (-0.988 - 0.151i)T \)
83 \( 1 + (0.610 - 0.791i)T \)
89 \( 1 + (-0.995 - 0.0950i)T \)
97 \( 1 + (0.998 + 0.0570i)T \)
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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

−18.39766065480250420309071331608, −17.69263588207685561656562568790, −16.76258428130327239156415951907, −16.20824939234983658107768148236, −15.66932463214581899231362955003, −14.97161026172148183442010519134, −14.315190455684575929432557007918, −13.779898832920354872798854856638, −12.996788150685525707822129289225, −11.98055586628990080199562617142, −11.18181108666315358437009838705, −10.54537932769774440978505465892, −9.73693478932058368178833688206, −9.24249299689659678030371582947, −8.55916110052683847643792076106, −8.05126989106122766119444139593, −7.23252605646735307370965044486, −6.60494075468943161205987843434, −5.85711606999993423374956565311, −4.71307002139482928368839909299, −4.13860201415233806914458827417, −2.97008469990592186550663033634, −2.320805694847169816141860006002, −1.55235091136251679761680864857, −0.59152850813199639821304530610, 0.60697889909331270950754150679, 1.47280902527653180038521328376, 2.16426871413443552593593224844, 2.87560963632705628343462982861, 3.91117223548179944372592781302, 4.047793902163466167736476760398, 5.80654676015735235562675858776, 6.31626578866426392573700988937, 7.517522604467521702248787335773, 7.70183753276234165063212965192, 8.659108132832233288816222905850, 8.93196533524278741148408652129, 9.880209262293371413261499644884, 10.48334109074891765630180748964, 11.03373338226105453851660982716, 12.11222737376176437161931833376, 12.670965630253030056747456921847, 13.224554457381523766173704827874, 14.02329927666707653454204456858, 14.90517069567407026929625768457, 15.555802145921561650893922357419, 15.977061501380247456974257765478, 17.10453641388426096562204832707, 17.48673793458591630152057007839, 18.370431866911021920538320328321

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