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Properties

Label 2-435-87.17-c1-0-33
Degree $2$
Conductor $435$
Sign $-0.154 - 0.987i$
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  + (−1.88 − 1.88i)2-s + (1.11 − 1.32i)3-s + 5.12i·4-s − 5-s + (−4.60 + 0.401i)6-s − 1.70·7-s + (5.90 − 5.90i)8-s + (−0.519 − 2.95i)9-s + (1.88 + 1.88i)10-s + (−2.51 − 2.51i)11-s + (6.80 + 5.71i)12-s − 4.77i·13-s + (3.22 + 3.22i)14-s + (−1.11 + 1.32i)15-s − 12.0·16-s + (2.45 + 2.45i)17-s + ⋯
L(s)  = 1  + (−1.33 − 1.33i)2-s + (0.642 − 0.765i)3-s + 2.56i·4-s − 0.447·5-s + (−1.88 + 0.164i)6-s − 0.645·7-s + (2.08 − 2.08i)8-s + (−0.173 − 0.984i)9-s + (0.596 + 0.596i)10-s + (−0.758 − 0.758i)11-s + (1.96 + 1.64i)12-s − 1.32i·13-s + (0.861 + 0.861i)14-s + (−0.287 + 0.342i)15-s − 3.00·16-s + (0.596 + 0.596i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $-0.154 - 0.987i$
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: \((2,\ 435,\ (\ :1/2),\ -0.154 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.161050 + 0.188217i\)
\(L(\frac12)\) \(\approx\) \(0.161050 + 0.188217i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.11 + 1.32i)T \)
5 \( 1 + T \)
29 \( 1 + (4.65 + 2.70i)T \)
good2 \( 1 + (1.88 + 1.88i)T + 2iT^{2} \)
7 \( 1 + 1.70T + 7T^{2} \)
11 \( 1 + (2.51 + 2.51i)T + 11iT^{2} \)
13 \( 1 + 4.77iT - 13T^{2} \)
17 \( 1 + (-2.45 - 2.45i)T + 17iT^{2} \)
19 \( 1 + (4.89 - 4.89i)T - 19iT^{2} \)
23 \( 1 - 6.05iT - 23T^{2} \)
31 \( 1 + (2.55 - 2.55i)T - 31iT^{2} \)
37 \( 1 + (-0.391 - 0.391i)T + 37iT^{2} \)
41 \( 1 + (3.30 - 3.30i)T - 41iT^{2} \)
43 \( 1 + (-1.12 + 1.12i)T - 43iT^{2} \)
47 \( 1 + (2.95 - 2.95i)T - 47iT^{2} \)
53 \( 1 + 10.3iT - 53T^{2} \)
59 \( 1 - 3.24iT - 59T^{2} \)
61 \( 1 + (-4.06 + 4.06i)T - 61iT^{2} \)
67 \( 1 - 0.366iT - 67T^{2} \)
71 \( 1 - 6.94T + 71T^{2} \)
73 \( 1 + (11.5 + 11.5i)T + 73iT^{2} \)
79 \( 1 + (-4.18 + 4.18i)T - 79iT^{2} \)
83 \( 1 + 16.8iT - 83T^{2} \)
89 \( 1 + (5.67 + 5.67i)T + 89iT^{2} \)
97 \( 1 + (9.06 + 9.06i)T + 97iT^{2} \)
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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

−10.36785520764813323872611485111, −9.714313992804996753954488070706, −8.577937715198949859013486302444, −8.054948723505256377479835287623, −7.45929035032138105214167366477, −5.96786701313560982621348866279, −3.54909639368307748898796291425, −3.17989431168646574539609761236, −1.75751300033985337344943885970, −0.21441334468850276438057661463, 2.33494492893089940890184152334, 4.30421257222696380864463395532, 5.24887809978530266867989721132, 6.66456097067075044617848567422, 7.29005436465409055056285223837, 8.269678797867454303089108794526, 9.051570236043553891407061570845, 9.608511425749274062122530855028, 10.44390848772151247000850902721, 11.21564481499610027380663337984

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