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Properties

Label 2-435-145.144-c1-0-3
Degree $2$
Conductor $435$
Sign $0.362 - 0.931i$
Analytic cond. $3.47349$
Root an. cond. $1.86373$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.60·2-s + 3-s + 0.574·4-s + (−2.22 + 0.232i)5-s − 1.60·6-s + 1.04i·7-s + 2.28·8-s + 9-s + (3.56 − 0.373i)10-s − 5.71i·11-s + 0.574·12-s + 4.68i·13-s − 1.66i·14-s + (−2.22 + 0.232i)15-s − 4.81·16-s + 2.22·17-s + ⋯
L(s)  = 1  − 1.13·2-s + 0.577·3-s + 0.287·4-s + (−0.994 + 0.104i)5-s − 0.655·6-s + 0.393i·7-s + 0.808·8-s + 0.333·9-s + (1.12 − 0.118i)10-s − 1.72i·11-s + 0.165·12-s + 1.30i·13-s − 0.446i·14-s + (−0.574 + 0.0601i)15-s − 1.20·16-s + 0.540·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.362 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $0.362 - 0.931i$
Analytic conductor: \(3.47349\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :1/2),\ 0.362 - 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.538163 + 0.367960i\)
\(L(\frac12)\) \(\approx\) \(0.538163 + 0.367960i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + (2.22 - 0.232i)T \)
29 \( 1 + (2.46 - 4.78i)T \)
good2 \( 1 + 1.60T + 2T^{2} \)
7 \( 1 - 1.04iT - 7T^{2} \)
11 \( 1 + 5.71iT - 11T^{2} \)
13 \( 1 - 4.68iT - 13T^{2} \)
17 \( 1 - 2.22T + 17T^{2} \)
19 \( 1 - 6.45iT - 19T^{2} \)
23 \( 1 - 6.19iT - 23T^{2} \)
31 \( 1 - 2.73iT - 31T^{2} \)
37 \( 1 - 7.69T + 37T^{2} \)
41 \( 1 - 2.07iT - 41T^{2} \)
43 \( 1 - 0.0721T + 43T^{2} \)
47 \( 1 + 3.86T + 47T^{2} \)
53 \( 1 - 10.2iT - 53T^{2} \)
59 \( 1 - 9.67T + 59T^{2} \)
61 \( 1 + 5.68iT - 61T^{2} \)
67 \( 1 - 8.51iT - 67T^{2} \)
71 \( 1 + 9.26T + 71T^{2} \)
73 \( 1 + 9.65T + 73T^{2} \)
79 \( 1 - 1.03iT - 79T^{2} \)
83 \( 1 - 1.94iT - 83T^{2} \)
89 \( 1 + 16.4iT - 89T^{2} \)
97 \( 1 - 1.12T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.24869530433448777204430292965, −10.25275089126027312428162722165, −9.223086915442980229426353337768, −8.623520737728335700659191823041, −7.948424647321346184809625963853, −7.16009073077967466686187376537, −5.76880923190783678204074329229, −4.19036805946356723940319472916, −3.24047970887480679852681138463, −1.37797590424918681095652665613, 0.64093105730469483870008168632, 2.49059251847074004020124858044, 4.09080023013695440289806481023, 4.86811452509713996987268291629, 6.93191317814994660038322608926, 7.60516535799834749076755397305, 8.160836174850975606635247122796, 9.135854474936574864693556159445, 9.976667654668664173127068478770, 10.60488252500010425991590545513

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