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Properties

Label 1-4235-4235.1038-r1-0-0
Degree $1$
Conductor $4235$
Sign $-0.282 + 0.959i$
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  + (0.524 + 0.851i)2-s + (0.406 + 0.913i)3-s + (−0.449 + 0.893i)4-s + (−0.564 + 0.825i)6-s + (−0.996 + 0.0855i)8-s + (−0.669 + 0.743i)9-s + (−0.998 − 0.0475i)12-s + (−0.967 − 0.254i)13-s + (−0.595 − 0.803i)16-s + (−0.768 − 0.640i)17-s + (−0.983 − 0.179i)18-s + (0.969 + 0.244i)19-s + (0.814 + 0.580i)23-s + (−0.483 − 0.875i)24-s + (−0.290 − 0.956i)26-s + (−0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (0.524 + 0.851i)2-s + (0.406 + 0.913i)3-s + (−0.449 + 0.893i)4-s + (−0.564 + 0.825i)6-s + (−0.996 + 0.0855i)8-s + (−0.669 + 0.743i)9-s + (−0.998 − 0.0475i)12-s + (−0.967 − 0.254i)13-s + (−0.595 − 0.803i)16-s + (−0.768 − 0.640i)17-s + (−0.983 − 0.179i)18-s + (0.969 + 0.244i)19-s + (0.814 + 0.580i)23-s + (−0.483 − 0.875i)24-s + (−0.290 − 0.956i)26-s + (−0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.282 + 0.959i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1038, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ -0.282 + 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.397193923 + 1.867429691i\)
\(L(\frac12)\) \(\approx\) \(1.397193923 + 1.867429691i\)
\(L(1)\) \(\approx\) \(0.8975287139 + 0.9280974689i\)
\(L(1)\) \(\approx\) \(0.8975287139 + 0.9280974689i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.524 + 0.851i)T \)
3 \( 1 + (0.406 + 0.913i)T \)
13 \( 1 + (-0.967 - 0.254i)T \)
17 \( 1 + (-0.768 - 0.640i)T \)
19 \( 1 + (0.969 + 0.244i)T \)
23 \( 1 + (0.814 + 0.580i)T \)
29 \( 1 + (0.466 + 0.884i)T \)
31 \( 1 + (-0.935 + 0.353i)T \)
37 \( 1 + (-0.151 - 0.988i)T \)
41 \( 1 + (0.610 + 0.791i)T \)
43 \( 1 + (0.755 - 0.654i)T \)
47 \( 1 + (-0.983 + 0.179i)T \)
53 \( 1 + (-0.803 - 0.595i)T \)
59 \( 1 + (-0.380 - 0.924i)T \)
61 \( 1 + (0.879 - 0.475i)T \)
67 \( 1 + (-0.690 - 0.723i)T \)
71 \( 1 + (0.897 + 0.441i)T \)
73 \( 1 + (0.508 + 0.861i)T \)
79 \( 1 + (-0.123 - 0.992i)T \)
83 \( 1 + (0.676 - 0.736i)T \)
89 \( 1 + (0.786 - 0.618i)T \)
97 \( 1 + (-0.856 - 0.516i)T \)
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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

−18.15607028539247445002285466378, −17.58738813620376258648335517226, −16.88276663344296013179064993643, −15.72156969828393346181879033593, −14.928339329901567904244762845776, −14.528284079000284729784380446164, −13.70789370533315434959829594749, −13.24369736268949398210165333367, −12.53506870697236137803059706515, −12.02461552299980331086730237135, −11.32224591019203754350439746600, −10.65665930746374559869644098354, −9.63024269199314989328878895903, −9.20074113917284410963331622494, −8.3733906662460211030935526570, −7.51518597890341096110367252807, −6.73390151514448501969258791461, −6.08831557094653341894363881821, −5.198892454815092918192494652712, −4.44580922428336858509050582911, −3.57909761424262516813886325929, −2.697883491724663295163676423656, −2.26856323713950494510304874888, −1.34131265778159214096112330133, −0.57154215140175866479855789484, 0.40329021454406282842046346730, 2.03920689216975440062669525958, 3.021163610014546435274752177570, 3.42249964605957056854496139017, 4.41359883306534050825594215672, 5.09556724307811985708177522254, 5.40359135338503334849964456171, 6.52848831927914481881650092841, 7.36344731206376352405412622914, 7.811103287726451623957185413176, 8.81580359958959888413758281397, 9.29476644598789808708746970337, 9.88804815013348875392263637993, 10.96358401182488026090901186982, 11.52601774635547551131163787804, 12.50740110187842202073479752625, 13.07915059557798041758295499319, 14.01562227375982579353530047117, 14.38246384672480971942632677437, 14.98528977431799632675886006273, 15.86632334887259364644184474706, 16.029787056358467315396893693, 16.8662938430438605973308046991, 17.57022295959579766372967908532, 18.09201690247749590029272943452

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