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Properties

Label 2-435-87.17-c1-0-32
Degree $2$
Conductor $435$
Sign $0.0254 + 0.999i$
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  + (0.0236 + 0.0236i)2-s + (1.39 − 1.03i)3-s − 1.99i·4-s − 5-s + (0.0572 + 0.00855i)6-s + 3.42·7-s + (0.0944 − 0.0944i)8-s + (0.877 − 2.86i)9-s + (−0.0236 − 0.0236i)10-s + (−4.10 − 4.10i)11-s + (−2.05 − 2.78i)12-s + 3.19i·13-s + (0.0809 + 0.0809i)14-s + (−1.39 + 1.03i)15-s − 3.99·16-s + (4.08 + 4.08i)17-s + ⋯
L(s)  = 1  + (0.0167 + 0.0167i)2-s + (0.803 − 0.594i)3-s − 0.999i·4-s − 0.447·5-s + (0.0233 + 0.00349i)6-s + 1.29·7-s + (0.0334 − 0.0334i)8-s + (0.292 − 0.956i)9-s + (−0.00747 − 0.00747i)10-s + (−1.23 − 1.23i)11-s + (−0.594 − 0.803i)12-s + 0.886i·13-s + (0.0216 + 0.0216i)14-s + (−0.359 + 0.266i)15-s − 0.998·16-s + (0.991 + 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0254 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0254 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $0.0254 + 0.999i$
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: \((2,\ 435,\ (\ :1/2),\ 0.0254 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27994 - 1.24771i\)
\(L(\frac12)\) \(\approx\) \(1.27994 - 1.24771i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.39 + 1.03i)T \)
5 \( 1 + T \)
29 \( 1 + (-4.91 - 2.20i)T \)
good2 \( 1 + (-0.0236 - 0.0236i)T + 2iT^{2} \)
7 \( 1 - 3.42T + 7T^{2} \)
11 \( 1 + (4.10 + 4.10i)T + 11iT^{2} \)
13 \( 1 - 3.19iT - 13T^{2} \)
17 \( 1 + (-4.08 - 4.08i)T + 17iT^{2} \)
19 \( 1 + (1.51 - 1.51i)T - 19iT^{2} \)
23 \( 1 + 3.49iT - 23T^{2} \)
31 \( 1 + (0.896 - 0.896i)T - 31iT^{2} \)
37 \( 1 + (1.53 + 1.53i)T + 37iT^{2} \)
41 \( 1 + (-4.78 + 4.78i)T - 41iT^{2} \)
43 \( 1 + (2.79 - 2.79i)T - 43iT^{2} \)
47 \( 1 + (-3.94 + 3.94i)T - 47iT^{2} \)
53 \( 1 + 0.219iT - 53T^{2} \)
59 \( 1 - 14.4iT - 59T^{2} \)
61 \( 1 + (-7.78 + 7.78i)T - 61iT^{2} \)
67 \( 1 - 11.2iT - 67T^{2} \)
71 \( 1 + 9.21T + 71T^{2} \)
73 \( 1 + (-10.7 - 10.7i)T + 73iT^{2} \)
79 \( 1 + (-1.49 + 1.49i)T - 79iT^{2} \)
83 \( 1 - 1.76iT - 83T^{2} \)
89 \( 1 + (-6.11 - 6.11i)T + 89iT^{2} \)
97 \( 1 + (5.10 + 5.10i)T + 97iT^{2} \)
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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

−10.84535247001714738357918066167, −10.18807492218347895753598785646, −8.714103004388807454009552751520, −8.342807998924989513783188531996, −7.38773267118106232231392313357, −6.19405814086199974213811061445, −5.20788547953232194174530409715, −3.96911021512214635763628011280, −2.41151709122145508756592823456, −1.15469390156596609730449845957, 2.30755848805507095047063974169, 3.28477833901139002699005209096, 4.60381246969465014438343304023, 5.08047621668424795019586620121, 7.39547770995674206698928546158, 7.78083408612144466835026009229, 8.341958926279087772042391396612, 9.523260654606446791907346061436, 10.45011887061128635348232343316, 11.34011938489439266124600458529

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