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Properties

Label 2-435-87.17-c1-0-37
Degree $2$
Conductor $435$
Sign $-0.907 + 0.419i$
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  + (−0.398 − 0.398i)2-s + (1.60 − 0.654i)3-s − 1.68i·4-s − 5-s + (−0.899 − 0.378i)6-s − 4.31·7-s + (−1.46 + 1.46i)8-s + (2.14 − 2.10i)9-s + (0.398 + 0.398i)10-s + (0.656 + 0.656i)11-s + (−1.10 − 2.69i)12-s − 2.82i·13-s + (1.72 + 1.72i)14-s + (−1.60 + 0.654i)15-s − 2.19·16-s + (−4.74 − 4.74i)17-s + ⋯
L(s)  = 1  + (−0.281 − 0.281i)2-s + (0.925 − 0.378i)3-s − 0.841i·4-s − 0.447·5-s + (−0.367 − 0.154i)6-s − 1.63·7-s + (−0.518 + 0.518i)8-s + (0.714 − 0.700i)9-s + (0.126 + 0.126i)10-s + (0.197 + 0.197i)11-s + (−0.318 − 0.778i)12-s − 0.782i·13-s + (0.459 + 0.459i)14-s + (−0.414 + 0.169i)15-s − 0.548·16-s + (−1.15 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $-0.907 + 0.419i$
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: \((2,\ 435,\ (\ :1/2),\ -0.907 + 0.419i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.206488 - 0.939530i\)
\(L(\frac12)\) \(\approx\) \(0.206488 - 0.939530i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.60 + 0.654i)T \)
5 \( 1 + T \)
29 \( 1 + (-1.25 - 5.23i)T \)
good2 \( 1 + (0.398 + 0.398i)T + 2iT^{2} \)
7 \( 1 + 4.31T + 7T^{2} \)
11 \( 1 + (-0.656 - 0.656i)T + 11iT^{2} \)
13 \( 1 + 2.82iT - 13T^{2} \)
17 \( 1 + (4.74 + 4.74i)T + 17iT^{2} \)
19 \( 1 + (-1.21 + 1.21i)T - 19iT^{2} \)
23 \( 1 + 1.96iT - 23T^{2} \)
31 \( 1 + (-2.91 + 2.91i)T - 31iT^{2} \)
37 \( 1 + (-3.89 - 3.89i)T + 37iT^{2} \)
41 \( 1 + (2.91 - 2.91i)T - 41iT^{2} \)
43 \( 1 + (-4.82 + 4.82i)T - 43iT^{2} \)
47 \( 1 + (6.22 - 6.22i)T - 47iT^{2} \)
53 \( 1 + 7.97iT - 53T^{2} \)
59 \( 1 + 7.91iT - 59T^{2} \)
61 \( 1 + (-2.74 + 2.74i)T - 61iT^{2} \)
67 \( 1 + 9.91iT - 67T^{2} \)
71 \( 1 - 5.70T + 71T^{2} \)
73 \( 1 + (-6.98 - 6.98i)T + 73iT^{2} \)
79 \( 1 + (-3.13 + 3.13i)T - 79iT^{2} \)
83 \( 1 + 13.0iT - 83T^{2} \)
89 \( 1 + (-10.5 - 10.5i)T + 89iT^{2} \)
97 \( 1 + (-8.43 - 8.43i)T + 97iT^{2} \)
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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

−10.54274565737729405946614624313, −9.586075396366748543401558962089, −9.264862823963704990858305669188, −8.169441458158182282392680641084, −6.88836444394377207711988958775, −6.39841912757927267486063808855, −4.83333617533309138276278790309, −3.34977790511709480759962802085, −2.48001595111949212245978587177, −0.57653764907458312612450611667, 2.56944487035404553322893166443, 3.64875092992432578470904675151, 4.20555639468303337190024256276, 6.28067519112364642074708171616, 7.01229647705078325842356520437, 8.009626494648711770495081116960, 8.870306130735585470384664323679, 9.413949458518641815189666863169, 10.35756335658400694111719373736, 11.59384941028493842808588311835

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