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Properties

Label 2-435-87.17-c1-0-36
Degree $2$
Conductor $435$
Sign $-0.440 + 0.897i$
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  + (1.34 + 1.34i)2-s + (−1.26 − 1.17i)3-s + 1.61i·4-s − 5-s + (−0.118 − 3.29i)6-s − 4.79·7-s + (0.514 − 0.514i)8-s + (0.215 + 2.99i)9-s + (−1.34 − 1.34i)10-s + (−3.27 − 3.27i)11-s + (1.90 − 2.05i)12-s − 2.36i·13-s + (−6.44 − 6.44i)14-s + (1.26 + 1.17i)15-s + 4.61·16-s + (−0.552 − 0.552i)17-s + ⋯
L(s)  = 1  + (0.950 + 0.950i)2-s + (−0.732 − 0.681i)3-s + 0.808i·4-s − 0.447·5-s + (−0.0484 − 1.34i)6-s − 1.81·7-s + (0.181 − 0.181i)8-s + (0.0719 + 0.997i)9-s + (−0.425 − 0.425i)10-s + (−0.987 − 0.987i)11-s + (0.550 − 0.592i)12-s − 0.656i·13-s + (−1.72 − 1.72i)14-s + (0.327 + 0.304i)15-s + 1.15·16-s + (−0.133 − 0.133i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.440 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $-0.440 + 0.897i$
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: \((2,\ 435,\ (\ :1/2),\ -0.440 + 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.242073 - 0.388613i\)
\(L(\frac12)\) \(\approx\) \(0.242073 - 0.388613i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.26 + 1.17i)T \)
5 \( 1 + T \)
29 \( 1 + (4.76 - 2.50i)T \)
good2 \( 1 + (-1.34 - 1.34i)T + 2iT^{2} \)
7 \( 1 + 4.79T + 7T^{2} \)
11 \( 1 + (3.27 + 3.27i)T + 11iT^{2} \)
13 \( 1 + 2.36iT - 13T^{2} \)
17 \( 1 + (0.552 + 0.552i)T + 17iT^{2} \)
19 \( 1 + (4.82 - 4.82i)T - 19iT^{2} \)
23 \( 1 + 7.39iT - 23T^{2} \)
31 \( 1 + (4.65 - 4.65i)T - 31iT^{2} \)
37 \( 1 + (-1.03 - 1.03i)T + 37iT^{2} \)
41 \( 1 + (-5.37 + 5.37i)T - 41iT^{2} \)
43 \( 1 + (-3.84 + 3.84i)T - 43iT^{2} \)
47 \( 1 + (-0.121 + 0.121i)T - 47iT^{2} \)
53 \( 1 - 4.08iT - 53T^{2} \)
59 \( 1 - 5.92iT - 59T^{2} \)
61 \( 1 + (2.25 - 2.25i)T - 61iT^{2} \)
67 \( 1 - 1.82iT - 67T^{2} \)
71 \( 1 + 8.68T + 71T^{2} \)
73 \( 1 + (-1.65 - 1.65i)T + 73iT^{2} \)
79 \( 1 + (-6.72 + 6.72i)T - 79iT^{2} \)
83 \( 1 + 7.82iT - 83T^{2} \)
89 \( 1 + (7.52 + 7.52i)T + 89iT^{2} \)
97 \( 1 + (10.4 + 10.4i)T + 97iT^{2} \)
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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

−10.66640347755867861716512717331, −10.40625747476945802565954028019, −8.690596897113657491531356213954, −7.63746501061668121880036095964, −6.84902803029220408986664718263, −6.00046055946605237437990044597, −5.56025842491438682277771735480, −4.12113103397333754793489426606, −2.94338413009402089002509158635, −0.21709327640877370370649593525, 2.49678384753511255986315437773, 3.66207898331545956095697181780, 4.34561289411653848451599713622, 5.42377222726649594518892711253, 6.46475549823668499074067907140, 7.52879847293383186040139320534, 9.341954004838263866447571934534, 9.794807029960700654414868023898, 10.87006461267220039223438986139, 11.36385284392771844547350290723

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