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Properties

Label 2-435-29.7-c1-0-2
Degree $2$
Conductor $435$
Sign $-0.977 + 0.209i$
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  + (0.412 + 1.80i)2-s + (−0.900 − 0.433i)3-s + (−1.29 + 0.625i)4-s + (0.222 + 0.974i)5-s + (0.412 − 1.80i)6-s + (−2.78 − 1.34i)7-s + (0.645 + 0.809i)8-s + (0.623 + 0.781i)9-s + (−1.67 + 0.804i)10-s + (−1.04 + 1.31i)11-s + 1.44·12-s + (−4.34 + 5.44i)13-s + (1.27 − 5.60i)14-s + (0.222 − 0.974i)15-s + (−2.99 + 3.75i)16-s − 2.75·17-s + ⋯
L(s)  = 1  + (0.291 + 1.27i)2-s + (−0.520 − 0.250i)3-s + (−0.649 + 0.312i)4-s + (0.0995 + 0.436i)5-s + (0.168 − 0.738i)6-s + (−1.05 − 0.507i)7-s + (0.228 + 0.286i)8-s + (0.207 + 0.260i)9-s + (−0.528 + 0.254i)10-s + (−0.316 + 0.396i)11-s + 0.416·12-s + (−1.20 + 1.51i)13-s + (0.341 − 1.49i)14-s + (0.0574 − 0.251i)15-s + (−0.748 + 0.939i)16-s − 0.667·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $-0.977 + 0.209i$
Analytic conductor: \(3.47349\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :1/2),\ -0.977 + 0.209i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0872512 - 0.825058i\)
\(L(\frac12)\) \(\approx\) \(0.0872512 - 0.825058i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.900 + 0.433i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
29 \( 1 + (-5.26 + 1.14i)T \)
good2 \( 1 + (-0.412 - 1.80i)T + (-1.80 + 0.867i)T^{2} \)
7 \( 1 + (2.78 + 1.34i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (1.04 - 1.31i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (4.34 - 5.44i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + 2.75T + 17T^{2} \)
19 \( 1 + (-0.370 + 0.178i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (-0.230 + 1.00i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (-1.18 - 5.18i)T + (-27.9 + 13.4i)T^{2} \)
37 \( 1 + (0.609 + 0.764i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + 2.72T + 41T^{2} \)
43 \( 1 + (-1.18 + 5.18i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-2.28 + 2.86i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-2.46 - 10.7i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 - 0.766T + 59T^{2} \)
61 \( 1 + (-9.96 - 4.79i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (1.18 + 1.48i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (-4.45 + 5.58i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (3.60 - 15.8i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + (-3.51 - 4.41i)T + (-17.5 + 77.0i)T^{2} \)
83 \( 1 + (-8.93 + 4.30i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (1.17 + 5.13i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (7.56 - 3.64i)T + (60.4 - 75.8i)T^{2} \)
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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

−11.73331914381092279911070259041, −10.61770493018053490295794606580, −9.843126653071107032486382074945, −8.728756807108625970508610358301, −7.32805087838512960647138331247, −6.91085756529566966229870800465, −6.34178195720168060059746219442, −5.10974293566735202912674288779, −4.20040379958076696136839576379, −2.34402088588052324053978815715, 0.47823968367375775903247101043, 2.48222550322781498234772879359, 3.34841393730486730376770677846, 4.69274427342329548777880379311, 5.58356391024381565873503554972, 6.73715111821151005185241111576, 8.053218950915987365344504450454, 9.394863567503954433351038970113, 9.968211684467723635757569981115, 10.67776797276682000361960061320

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