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Properties

Label 2-435-87.41-c1-0-9
Degree $2$
Conductor $435$
Sign $-0.261 - 0.965i$
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  + (−0.203 + 0.203i)2-s + (−1.68 + 0.407i)3-s + 1.91i·4-s − 5-s + (0.259 − 0.424i)6-s + 4.71·7-s + (−0.795 − 0.795i)8-s + (2.66 − 1.37i)9-s + (0.203 − 0.203i)10-s + (1.95 − 1.95i)11-s + (−0.781 − 3.22i)12-s + 5.24i·13-s + (−0.958 + 0.958i)14-s + (1.68 − 0.407i)15-s − 3.51·16-s + (−2.21 + 2.21i)17-s + ⋯
L(s)  = 1  + (−0.143 + 0.143i)2-s + (−0.971 + 0.235i)3-s + 0.958i·4-s − 0.447·5-s + (0.105 − 0.173i)6-s + 1.78·7-s + (−0.281 − 0.281i)8-s + (0.889 − 0.457i)9-s + (0.0642 − 0.0642i)10-s + (0.589 − 0.589i)11-s + (−0.225 − 0.931i)12-s + 1.45i·13-s + (−0.256 + 0.256i)14-s + (0.434 − 0.105i)15-s − 0.877·16-s + (−0.536 + 0.536i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.261 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.261 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $-0.261 - 0.965i$
Analytic conductor: \(3.47349\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :1/2),\ -0.261 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.591649 + 0.773400i\)
\(L(\frac12)\) \(\approx\) \(0.591649 + 0.773400i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.68 - 0.407i)T \)
5 \( 1 + T \)
29 \( 1 + (5.31 + 0.873i)T \)
good2 \( 1 + (0.203 - 0.203i)T - 2iT^{2} \)
7 \( 1 - 4.71T + 7T^{2} \)
11 \( 1 + (-1.95 + 1.95i)T - 11iT^{2} \)
13 \( 1 - 5.24iT - 13T^{2} \)
17 \( 1 + (2.21 - 2.21i)T - 17iT^{2} \)
19 \( 1 + (-0.0703 - 0.0703i)T + 19iT^{2} \)
23 \( 1 - 7.77iT - 23T^{2} \)
31 \( 1 + (0.672 + 0.672i)T + 31iT^{2} \)
37 \( 1 + (-2.42 + 2.42i)T - 37iT^{2} \)
41 \( 1 + (-6.09 - 6.09i)T + 41iT^{2} \)
43 \( 1 + (6.01 + 6.01i)T + 43iT^{2} \)
47 \( 1 + (-5.31 - 5.31i)T + 47iT^{2} \)
53 \( 1 + 4.43iT - 53T^{2} \)
59 \( 1 + 2.33iT - 59T^{2} \)
61 \( 1 + (2.26 + 2.26i)T + 61iT^{2} \)
67 \( 1 - 14.8iT - 67T^{2} \)
71 \( 1 - 3.36T + 71T^{2} \)
73 \( 1 + (8.05 - 8.05i)T - 73iT^{2} \)
79 \( 1 + (1.03 + 1.03i)T + 79iT^{2} \)
83 \( 1 - 8.25iT - 83T^{2} \)
89 \( 1 + (-3.11 + 3.11i)T - 89iT^{2} \)
97 \( 1 + (-3.99 + 3.99i)T - 97iT^{2} \)
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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

−11.48248771348130289547865341261, −11.04965423206349958673391922053, −9.429177564004300368770840012451, −8.627483144810426814255891012722, −7.66339307534016860615809121699, −6.92173056061162272860644202136, −5.67825815819163644730192177591, −4.42092171133233864557100624996, −3.90534919412541090278062478917, −1.67468615993098656446890873256, 0.799330319166515320219925873483, 2.06809951921887781213474651134, 4.48078333077278661456120031572, 5.02452246297979349578344678037, 6.00180355297988228639348416224, 7.16456060030274369747353002440, 8.024025874340952271639395465749, 9.128575530776128342680825445027, 10.47553171312465052650361680501, 10.79740124163619230245260173548

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