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Properties

Label 1-4235-4235.1497-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.576 - 0.816i$
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  + (−0.989 − 0.142i)2-s + i·3-s + (0.959 + 0.281i)4-s + (0.142 − 0.989i)6-s + (−0.909 − 0.415i)8-s − 9-s + (−0.281 + 0.959i)12-s + (0.281 − 0.959i)13-s + (0.841 + 0.540i)16-s + (0.755 + 0.654i)17-s + (0.989 + 0.142i)18-s + (−0.654 − 0.755i)19-s + (0.540 − 0.841i)23-s + (0.415 − 0.909i)24-s + (−0.415 + 0.909i)26-s − i·27-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)2-s + i·3-s + (0.959 + 0.281i)4-s + (0.142 − 0.989i)6-s + (−0.909 − 0.415i)8-s − 9-s + (−0.281 + 0.959i)12-s + (0.281 − 0.959i)13-s + (0.841 + 0.540i)16-s + (0.755 + 0.654i)17-s + (0.989 + 0.142i)18-s + (−0.654 − 0.755i)19-s + (0.540 − 0.841i)23-s + (0.415 − 0.909i)24-s + (−0.415 + 0.909i)26-s − i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.576 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.576 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.576 - 0.816i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.576 - 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1275566366 - 0.2462114409i\)
\(L(\frac12)\) \(\approx\) \(0.1275566366 - 0.2462114409i\)
\(L(1)\) \(\approx\) \(0.6019748789 + 0.07476969004i\)
\(L(1)\) \(\approx\) \(0.6019748789 + 0.07476969004i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.989 - 0.142i)T \)
3 \( 1 + iT \)
13 \( 1 + (0.281 - 0.959i)T \)
17 \( 1 + (0.755 + 0.654i)T \)
19 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (0.540 - 0.841i)T \)
29 \( 1 + (0.654 + 0.755i)T \)
31 \( 1 + (0.959 - 0.281i)T \)
37 \( 1 + (-0.281 - 0.959i)T \)
41 \( 1 + (0.142 - 0.989i)T \)
43 \( 1 + (-0.909 - 0.415i)T \)
47 \( 1 + (-0.989 + 0.142i)T \)
53 \( 1 + (-0.540 - 0.841i)T \)
59 \( 1 + (-0.142 - 0.989i)T \)
61 \( 1 + (0.142 + 0.989i)T \)
67 \( 1 + (-0.989 - 0.142i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (-0.540 + 0.841i)T \)
79 \( 1 + (-0.415 - 0.909i)T \)
83 \( 1 + (0.540 + 0.841i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (-0.909 - 0.415i)T \)
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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

−18.63478906427906226971156997705, −18.02982437941297975338805646534, −17.305468066428167516434867355492, −16.75932071651665457489591112092, −16.19442816187698241732414357408, −15.27112048897193961628711910162, −14.564296747904130520582580869939, −13.88408589033080870198337662237, −13.2009362655149732550749003612, −12.183075047005134421590429921545, −11.75046151560816153913317461069, −11.22707119947732273699503722494, −10.21951381587543738339199447289, −9.60670816144627151143396488677, −8.78493057476312498944135087845, −8.15061597801527256059206001700, −7.6245224573822004444295489747, −6.75255208997347721868155007852, −6.35628555591280122451801124344, −5.582557993203110230501604989255, −4.57162350629259949204218617365, −3.21896572776763338204812337777, −2.68211495140958456890443609654, −1.50954280031908607508528523649, −1.306060670406436482861419088104, 0.114020073955443073427860113652, 1.143062360283938478420644739051, 2.328842245565984059436609956245, 3.05377218262555252084474206324, 3.67487067947575890479513346252, 4.68692669358986232079076548598, 5.5184714348534863441681430333, 6.27339572300670460066993326069, 7.03165405761948852076589242583, 8.14063573769124081235977938856, 8.45174625029502985674634745886, 9.179727075805050521375400692010, 9.96624426886788146404621581507, 10.60458385707317070344634736587, 10.839268735985152677217216927645, 11.81366632201497222519481340769, 12.485063349108522525420014647694, 13.26730065529450208383432793954, 14.43418032342312087079407168430, 14.9676435379952808172958399915, 15.56209997352390550989045731591, 16.21729560902597349663461358068, 16.78106728441815919681426838846, 17.54939534977130043561970654414, 17.846532796577728615425575479447

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