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Properties

Label 2-435-87.17-c1-0-3
Degree $2$
Conductor $435$
Sign $0.680 + 0.733i$
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  + (−1.48 − 1.48i)2-s + (−1.36 − 1.06i)3-s + 2.43i·4-s − 5-s + (0.442 + 3.61i)6-s − 1.31·7-s + (0.641 − 0.641i)8-s + (0.722 + 2.91i)9-s + (1.48 + 1.48i)10-s + (1.11 + 1.11i)11-s + (2.59 − 3.31i)12-s − 0.320i·13-s + (1.95 + 1.95i)14-s + (1.36 + 1.06i)15-s + 2.95·16-s + (0.771 + 0.771i)17-s + ⋯
L(s)  = 1  + (−1.05 − 1.05i)2-s + (−0.787 − 0.616i)3-s + 1.21i·4-s − 0.447·5-s + (0.180 + 1.47i)6-s − 0.495·7-s + (0.226 − 0.226i)8-s + (0.240 + 0.970i)9-s + (0.470 + 0.470i)10-s + (0.335 + 0.335i)11-s + (0.748 − 0.957i)12-s − 0.0889i·13-s + (0.521 + 0.521i)14-s + (0.352 + 0.275i)15-s + 0.737·16-s + (0.187 + 0.187i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 + 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.680 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $0.680 + 0.733i$
Analytic conductor: \(3.47349\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :1/2),\ 0.680 + 0.733i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.399959 - 0.174497i\)
\(L(\frac12)\) \(\approx\) \(0.399959 - 0.174497i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.36 + 1.06i)T \)
5 \( 1 + T \)
29 \( 1 + (-0.168 - 5.38i)T \)
good2 \( 1 + (1.48 + 1.48i)T + 2iT^{2} \)
7 \( 1 + 1.31T + 7T^{2} \)
11 \( 1 + (-1.11 - 1.11i)T + 11iT^{2} \)
13 \( 1 + 0.320iT - 13T^{2} \)
17 \( 1 + (-0.771 - 0.771i)T + 17iT^{2} \)
19 \( 1 + (0.562 - 0.562i)T - 19iT^{2} \)
23 \( 1 - 1.90iT - 23T^{2} \)
31 \( 1 + (-3.49 + 3.49i)T - 31iT^{2} \)
37 \( 1 + (-0.254 - 0.254i)T + 37iT^{2} \)
41 \( 1 + (-0.109 + 0.109i)T - 41iT^{2} \)
43 \( 1 + (-3.91 + 3.91i)T - 43iT^{2} \)
47 \( 1 + (-5.63 + 5.63i)T - 47iT^{2} \)
53 \( 1 - 6.38iT - 53T^{2} \)
59 \( 1 + 1.03iT - 59T^{2} \)
61 \( 1 + (-4.99 + 4.99i)T - 61iT^{2} \)
67 \( 1 - 11.6iT - 67T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 + (-9.52 - 9.52i)T + 73iT^{2} \)
79 \( 1 + (2.05 - 2.05i)T - 79iT^{2} \)
83 \( 1 - 7.00iT - 83T^{2} \)
89 \( 1 + (-9.17 - 9.17i)T + 89iT^{2} \)
97 \( 1 + (-12.7 - 12.7i)T + 97iT^{2} \)
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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

−11.00176034004970483847244536196, −10.28689695831280127253165491397, −9.436925119715321190673758099760, −8.397034731394763267544243934218, −7.52544723665090371543662049473, −6.53103120698840672275236048061, −5.33712713902563494376740281136, −3.78061632158098811023586836836, −2.34367457863809190828622754121, −0.965252506768556545714564581588, 0.59177154996541453191102467639, 3.42207333922996409433069980393, 4.70003408437216877628501510838, 5.99741176434406665835994340154, 6.55241697779050255746690284012, 7.56035191936343516769876321145, 8.585177676078175682085043566278, 9.401736569831760691257947625890, 10.09242796393090462916498535005, 11.01537137133939779218708317244

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