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Properties

Label 1-4235-4235.1398-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.686 + 0.727i$
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  + (−0.458 + 0.888i)2-s + (0.866 + 0.5i)3-s + (−0.580 − 0.814i)4-s + (−0.841 + 0.540i)6-s + (0.989 − 0.142i)8-s + (0.5 + 0.866i)9-s + (−0.0950 − 0.995i)12-s + (−0.909 − 0.415i)13-s + (−0.327 + 0.945i)16-s + (0.690 + 0.723i)17-s + (−0.998 + 0.0475i)18-s + (0.723 + 0.690i)19-s + (−0.945 − 0.327i)23-s + (0.928 + 0.371i)24-s + (0.786 − 0.618i)26-s + i·27-s + ⋯
L(s)  = 1  + (−0.458 + 0.888i)2-s + (0.866 + 0.5i)3-s + (−0.580 − 0.814i)4-s + (−0.841 + 0.540i)6-s + (0.989 − 0.142i)8-s + (0.5 + 0.866i)9-s + (−0.0950 − 0.995i)12-s + (−0.909 − 0.415i)13-s + (−0.327 + 0.945i)16-s + (0.690 + 0.723i)17-s + (−0.998 + 0.0475i)18-s + (0.723 + 0.690i)19-s + (−0.945 − 0.327i)23-s + (0.928 + 0.371i)24-s + (0.786 − 0.618i)26-s + i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.686 + 0.727i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1398, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.686 + 0.727i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6829314885 + 1.583250328i\)
\(L(\frac12)\) \(\approx\) \(0.6829314885 + 1.583250328i\)
\(L(1)\) \(\approx\) \(0.8891604789 + 0.6505017673i\)
\(L(1)\) \(\approx\) \(0.8891604789 + 0.6505017673i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.458 + 0.888i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.909 - 0.415i)T \)
17 \( 1 + (0.690 + 0.723i)T \)
19 \( 1 + (0.723 + 0.690i)T \)
23 \( 1 + (-0.945 - 0.327i)T \)
29 \( 1 + (0.959 - 0.281i)T \)
31 \( 1 + (0.995 + 0.0950i)T \)
37 \( 1 + (-0.814 - 0.580i)T \)
41 \( 1 + (-0.841 + 0.540i)T \)
43 \( 1 + (0.989 - 0.142i)T \)
47 \( 1 + (0.998 + 0.0475i)T \)
53 \( 1 + (0.945 - 0.327i)T \)
59 \( 1 + (-0.888 + 0.458i)T \)
61 \( 1 + (-0.0475 + 0.998i)T \)
67 \( 1 + (0.998 - 0.0475i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (-0.189 - 0.981i)T \)
79 \( 1 + (-0.928 + 0.371i)T \)
83 \( 1 + (0.755 + 0.654i)T \)
89 \( 1 + (0.235 + 0.971i)T \)
97 \( 1 + (0.989 - 0.142i)T \)
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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

−18.40392638299662330425415567961, −17.52652688013733664808592700254, −17.167924615665221629606912497882, −16.021421095276631427679454360105, −15.53204158665754342135802179587, −14.31555477120093817462827875011, −13.98206005275702973581218635084, −13.42919331583832164563228676808, −12.44963703834174674051923786055, −12.01641622984494426139132962371, −11.53272839693563554881148196594, −10.24706258632831063831436451891, −9.92799946265557369777735507450, −9.12676240655476617805354843271, −8.62686461771412491672327059877, −7.66430985445583345911720879843, −7.370039003295703643200225845263, −6.4778020327326017343888221431, −5.18270714772733992486831817759, −4.4651052638647673886918834812, −3.5661297694681636135594604016, −2.860540409873883998957053350162, −2.31168339321247311207311420176, −1.42197288353834914353297227201, −0.578764022784411367546070859389, 0.98867933276835681729093178027, 1.96788190094036955813800044846, 2.87027441008109883208248274264, 3.850579579129611536286715900706, 4.50409216168114755186920219392, 5.340217399847700261304869749392, 5.96805112053769674400246762023, 6.99111048675737554345842226472, 7.711858854277595082202087735304, 8.15259330101665442134445224814, 8.827260372933342988901760507822, 9.65239796479798374560325376791, 10.26596293851238254199974032903, 10.47745814857616583165738645276, 11.92163616472043810267420226934, 12.57105981298175741271138801635, 13.632665904155758734806463345568, 14.0453145095747194816183197431, 14.6798332281056531370006812421, 15.2539834403494363786681018229, 15.900347394647514599843423477291, 16.47383819944530568209655939783, 17.18141584838607290221753260042, 17.85416706091995489360695120981, 18.65773200855303852770204727082

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