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Properties

Label 2-435-435.434-c0-0-3
Degree $2$
Conductor $435$
Sign $-0.707 + 0.707i$
Analytic cond. $0.217093$
Root an. cond. $0.465932$
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  − 1.41i·2-s + (−0.707 + 0.707i)3-s − 1.00·4-s − i·5-s + (1.00 + 1.00i)6-s − 1.00i·9-s − 1.41·10-s + (0.707 − 0.707i)12-s + (0.707 + 0.707i)15-s − 0.999·16-s − 1.41i·17-s − 1.41·18-s + 1.00i·20-s − 25-s + (0.707 + 0.707i)27-s + ⋯
L(s)  = 1  − 1.41i·2-s + (−0.707 + 0.707i)3-s − 1.00·4-s − i·5-s + (1.00 + 1.00i)6-s − 1.00i·9-s − 1.41·10-s + (0.707 − 0.707i)12-s + (0.707 + 0.707i)15-s − 0.999·16-s − 1.41i·17-s − 1.41·18-s + 1.00i·20-s − 25-s + (0.707 + 0.707i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(0.217093\)
Root analytic conductor: \(0.465932\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (434, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :0),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6549346783\)
\(L(\frac12)\) \(\approx\) \(0.6549346783\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + iT \)
29 \( 1 - iT \)
good2 \( 1 + 1.41iT - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.41T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 1.41T + T^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 2iT - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 2iT - T^{2} \)
73 \( 1 + 1.41T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 1.41T + T^{2} \)
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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

−11.10925556937784685257595298515, −10.30107898932773866462678532612, −9.355172065686950677158247970746, −9.016485434287188897409415824302, −7.37611956487662343087107882024, −5.93876239627619688569353930404, −4.83169075143660255869303250892, −4.14429183287958096912215365830, −2.82987199154408230938686832926, −1.02405564282509788393045643942, 2.29200787907349334667327287820, 4.20323200289694540433527767246, 5.64776528050153626290529813582, 6.16702982690644972467450122874, 6.97099743013341101755526103891, 7.71690287397927697091285302851, 8.458656965702244809334768731076, 9.932589286928448295718445679514, 10.91973120262719832743795947517, 11.57612340855838048530728519379

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