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Properties

Label 1-4235-4235.867-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.988 - 0.149i$
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  + (0.931 − 0.362i)2-s + (0.951 + 0.309i)3-s + (0.736 − 0.676i)4-s + (0.998 − 0.0570i)6-s + (0.441 − 0.897i)8-s + (0.809 + 0.587i)9-s + (0.909 − 0.415i)12-s + (0.980 + 0.198i)13-s + (0.0855 − 0.996i)16-s + (−0.791 − 0.610i)17-s + (0.967 + 0.254i)18-s + (−0.564 + 0.825i)19-s + (0.755 + 0.654i)23-s + (0.696 − 0.717i)24-s + (0.985 − 0.170i)26-s + (0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (0.931 − 0.362i)2-s + (0.951 + 0.309i)3-s + (0.736 − 0.676i)4-s + (0.998 − 0.0570i)6-s + (0.441 − 0.897i)8-s + (0.809 + 0.587i)9-s + (0.909 − 0.415i)12-s + (0.980 + 0.198i)13-s + (0.0855 − 0.996i)16-s + (−0.791 − 0.610i)17-s + (0.967 + 0.254i)18-s + (−0.564 + 0.825i)19-s + (0.755 + 0.654i)23-s + (0.696 − 0.717i)24-s + (0.985 − 0.170i)26-s + (0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.988 - 0.149i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (867, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.988 - 0.149i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.323596059 - 0.4001231917i\)
\(L(\frac12)\) \(\approx\) \(5.323596059 - 0.4001231917i\)
\(L(1)\) \(\approx\) \(2.698929561 - 0.2885163137i\)
\(L(1)\) \(\approx\) \(2.698929561 - 0.2885163137i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.931 - 0.362i)T \)
3 \( 1 + (0.951 + 0.309i)T \)
13 \( 1 + (0.980 + 0.198i)T \)
17 \( 1 + (-0.791 - 0.610i)T \)
19 \( 1 + (-0.564 + 0.825i)T \)
23 \( 1 + (0.755 + 0.654i)T \)
29 \( 1 + (0.0285 + 0.999i)T \)
31 \( 1 + (0.870 - 0.491i)T \)
37 \( 1 + (-0.113 + 0.993i)T \)
41 \( 1 + (-0.774 + 0.633i)T \)
43 \( 1 + (0.989 + 0.142i)T \)
47 \( 1 + (-0.967 + 0.254i)T \)
53 \( 1 + (0.996 - 0.0855i)T \)
59 \( 1 + (0.774 + 0.633i)T \)
61 \( 1 + (0.362 - 0.931i)T \)
67 \( 1 + (-0.540 + 0.841i)T \)
71 \( 1 + (0.941 - 0.336i)T \)
73 \( 1 + (0.389 - 0.921i)T \)
79 \( 1 + (0.466 - 0.884i)T \)
83 \( 1 + (-0.226 + 0.974i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (-0.717 - 0.696i)T \)
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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

−18.38859269161653136910797074990, −17.62056493887100128985549862217, −17.028815536573797515623148198747, −15.972873515344495902730506561512, −15.509869399521537318067977891191, −15.013905193741246497581019214935, −14.26542243826259719499429403926, −13.60506873333065225177432232860, −13.09497477208920436249820280001, −12.63665252770775795266837220146, −11.72684271115859749289201532575, −10.937884847763699547181368712920, −10.30367315705299234667232922710, −9.08082664509619781656982635235, −8.52925594816143201857887364781, −8.03793722132547039000323980963, −6.98473425305389767850960547943, −6.64458985524331497291120961541, −5.84507912138921885280643385017, −4.82236351034409069572806046765, −4.07996643815394534964842068985, −3.556923356047161644774520542780, −2.56049007696910702444898612994, −2.145588196889830979448881099355, −0.98128170255358063249770290046, 1.180450748327754090741959211123, 1.848890420622450739448280132069, 2.76271136391553838721065571435, 3.351273235471891333566728788731, 4.093871369451675279821248806218, 4.70527331392245025757790358547, 5.50903848921208569450312032255, 6.499431726851632260866792962305, 7.01413308191671331339112385740, 8.01657221280003252091419468146, 8.694631126291331413239899795494, 9.51129733443991182714137305580, 10.15264596593418065652401634714, 10.939937080086735015783303714356, 11.45917495495361723449042834417, 12.40760043677028761851180233617, 13.15122608287874352972389161726, 13.6103552853525248356961550192, 14.137548034852905135235899224386, 15.06780545281707488168255798462, 15.275432924909239155638672249904, 16.18959567721001834338406424576, 16.589921798821167147493008299841, 17.86515131010924710688316470427, 18.664499308548087505726660757

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