Properties

Label 2-435-5.4-c1-0-5
Degree $2$
Conductor $435$
Sign $0.821 + 0.569i$
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  − 2.51i·2-s + i·3-s − 4.34·4-s + (−1.27 + 1.83i)5-s + 2.51·6-s − 0.173i·7-s + 5.90i·8-s − 9-s + (4.62 + 3.20i)10-s + 5.08·11-s − 4.34i·12-s + 1.82i·13-s − 0.436·14-s + (−1.83 − 1.27i)15-s + 6.19·16-s + 4.24i·17-s + ⋯
L(s)  = 1  − 1.78i·2-s + 0.577i·3-s − 2.17·4-s + (−0.569 + 0.821i)5-s + 1.02·6-s − 0.0655i·7-s + 2.08i·8-s − 0.333·9-s + (1.46 + 1.01i)10-s + 1.53·11-s − 1.25i·12-s + 0.506i·13-s − 0.116·14-s + (−0.474 − 0.328i)15-s + 1.54·16-s + 1.02i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.821 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $0.821 + 0.569i$
Analytic conductor: \(3.47349\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :1/2),\ 0.821 + 0.569i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07073 - 0.334901i\)
\(L(\frac12)\) \(\approx\) \(1.07073 - 0.334901i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 + (1.27 - 1.83i)T \)
29 \( 1 + T \)
good2 \( 1 + 2.51iT - 2T^{2} \)
7 \( 1 + 0.173iT - 7T^{2} \)
11 \( 1 - 5.08T + 11T^{2} \)
13 \( 1 - 1.82iT - 13T^{2} \)
17 \( 1 - 4.24iT - 17T^{2} \)
19 \( 1 - 8.62T + 19T^{2} \)
23 \( 1 - 3.16iT - 23T^{2} \)
31 \( 1 - 3.22T + 31T^{2} \)
37 \( 1 + 1.97iT - 37T^{2} \)
41 \( 1 + 9.96T + 41T^{2} \)
43 \( 1 - 7.91iT - 43T^{2} \)
47 \( 1 - 8.66iT - 47T^{2} \)
53 \( 1 - 5.40iT - 53T^{2} \)
59 \( 1 + 7.66T + 59T^{2} \)
61 \( 1 - 2.76T + 61T^{2} \)
67 \( 1 + 8.13iT - 67T^{2} \)
71 \( 1 + 6.25T + 71T^{2} \)
73 \( 1 + 6.74iT - 73T^{2} \)
79 \( 1 + 4.54T + 79T^{2} \)
83 \( 1 + 11.6iT - 83T^{2} \)
89 \( 1 - 16.4T + 89T^{2} \)
97 \( 1 + 2.74iT - 97T^{2} \)
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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

−11.20870354285811323313436612716, −10.30511666117055469018392475445, −9.564852368127007324699239570163, −8.884421099779445911222063760602, −7.57733197038064211091659495233, −6.19962614103983441353820108384, −4.62419097202977407753476878517, −3.73393973915889489529018765408, −3.12367358760084048251845704331, −1.46681786629662789874171379008, 0.851196706449243457577737574702, 3.63512508502656866723192111736, 4.89223943211723950137535025966, 5.61664760318209927021092219972, 6.82571383342698626744473048381, 7.34165493705698315603096436168, 8.331869401286255530983125415286, 8.954366826204978780472727740872, 9.775905051048762296206552857606, 11.72499743842598479169961935383

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