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Properties

Label 1-4235-4235.324-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.817 - 0.576i$
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  + (−0.749 + 0.662i)2-s + (0.978 + 0.207i)3-s + (0.123 − 0.992i)4-s + (−0.870 + 0.491i)6-s + (0.564 + 0.825i)8-s + (0.913 + 0.406i)9-s + (0.327 − 0.945i)12-s + (−0.974 − 0.226i)13-s + (−0.969 − 0.244i)16-s + (−0.988 − 0.151i)17-s + (−0.953 + 0.299i)18-s + (0.161 − 0.986i)19-s + (−0.928 + 0.371i)23-s + (0.380 + 0.924i)24-s + (0.879 − 0.475i)26-s + (0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (−0.749 + 0.662i)2-s + (0.978 + 0.207i)3-s + (0.123 − 0.992i)4-s + (−0.870 + 0.491i)6-s + (0.564 + 0.825i)8-s + (0.913 + 0.406i)9-s + (0.327 − 0.945i)12-s + (−0.974 − 0.226i)13-s + (−0.969 − 0.244i)16-s + (−0.988 − 0.151i)17-s + (−0.953 + 0.299i)18-s + (0.161 − 0.986i)19-s + (−0.928 + 0.371i)23-s + (0.380 + 0.924i)24-s + (0.879 − 0.475i)26-s + (0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.817 - 0.576i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.817 - 0.576i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.132102296 - 0.3590730573i\)
\(L(\frac12)\) \(\approx\) \(1.132102296 - 0.3590730573i\)
\(L(1)\) \(\approx\) \(0.9090836481 + 0.1771647193i\)
\(L(1)\) \(\approx\) \(0.9090836481 + 0.1771647193i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.749 + 0.662i)T \)
3 \( 1 + (0.978 + 0.207i)T \)
13 \( 1 + (-0.974 - 0.226i)T \)
17 \( 1 + (-0.988 - 0.151i)T \)
19 \( 1 + (0.161 - 0.986i)T \)
23 \( 1 + (-0.928 + 0.371i)T \)
29 \( 1 + (-0.254 - 0.967i)T \)
31 \( 1 + (0.820 + 0.572i)T \)
37 \( 1 + (-0.483 + 0.875i)T \)
41 \( 1 + (0.993 + 0.113i)T \)
43 \( 1 + (0.959 - 0.281i)T \)
47 \( 1 + (-0.953 - 0.299i)T \)
53 \( 1 + (0.969 - 0.244i)T \)
59 \( 1 + (-0.398 + 0.917i)T \)
61 \( 1 + (-0.948 + 0.318i)T \)
67 \( 1 + (-0.580 - 0.814i)T \)
71 \( 1 + (-0.998 - 0.0570i)T \)
73 \( 1 + (0.830 - 0.556i)T \)
79 \( 1 + (-0.761 - 0.647i)T \)
83 \( 1 + (0.466 - 0.884i)T \)
89 \( 1 + (0.0475 - 0.998i)T \)
97 \( 1 + (-0.610 + 0.791i)T \)
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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

−18.515265159458748079277022115362, −17.944650028760852067915720965499, −17.32402177458788691855409905766, −16.39961235659156853298663515689, −15.88747420734205267188857099307, −15.01784333495117608728800287799, −14.250414621512440311919753865035, −13.71662348304101947558349065500, −12.6456818357367688789160897069, −12.52478176909264988983066427785, −11.6088232916998848449484026663, −10.73820550133657090382256909109, −10.042227933353343557164421530985, −9.46721655876959654895733376012, −8.842137065814838148635420640053, −8.136500918031101478692246029583, −7.54015585137148023603836457891, −6.918014452015933335971566268937, −6.01717373090667749767398090958, −4.61312236162906903345085692814, −4.05745864880022735070998018392, −3.25685106918962646771659358803, −2.35362682874885417820906466174, −1.99718718706148519032519750804, −0.97291399568658202027609852547, 0.39087056679090461754421597120, 1.65117822154432315869908177402, 2.36704593965760863677985284743, 3.05935761369437084730966860786, 4.413441754718402770320518577546, 4.71925424105238235224114207738, 5.807735136728767625365080783920, 6.65631303190579354800404959291, 7.40717341861265770710735136580, 7.82212290673116901393772051144, 8.74287289943168245313160676151, 9.13823056594015730970304256636, 9.94460014676461507239770722148, 10.35873167031563423591268850104, 11.309478996270188979839159180175, 12.1027948125287389685153692104, 13.23375381451183788836808555520, 13.71280331885407526488694418331, 14.36007226724694224401276039440, 15.243143331461799090188505982, 15.422224943071820748106078084360, 16.169698090683050203130646634937, 16.959909603279388584794640114330, 17.77153010429073094186517067127, 18.09504460878849457367211536730

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