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Properties

Label 2-435-87.41-c1-0-10
Degree $2$
Conductor $435$
Sign $-0.336 - 0.941i$
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 + (0.654 + 1.60i)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 + (−2.69 + 1.10i)12-s + 2.82i·13-s + (−1.72 + 1.72i)14-s + (0.654 + 1.60i)15-s − 2.19·16-s + (4.74 − 4.74i)17-s + ⋯
L(s)  = 1  + (0.281 − 0.281i)2-s + (0.378 + 0.925i)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 + (−0.778 + 0.318i)12-s + 0.782i·13-s + (−0.459 + 0.459i)14-s + (0.169 + 0.414i)15-s − 0.548·16-s + (1.15 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $-0.336 - 0.941i$
Analytic conductor: \(3.47349\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :1/2),\ -0.336 - 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.881892 + 1.25111i\)
\(L(\frac12)\) \(\approx\) \(0.881892 + 1.25111i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.654 - 1.60i)T \)
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

−11.43369528289542930444093604999, −10.36848872753445487063562708527, −9.544921667036682928947887488191, −9.072049097023514517877058488349, −7.80018444148363812087034570722, −6.79145981111858771673906386953, −5.51937311264141469727408975138, −4.36173938305222711210261019413, −3.29942064907982145089235254767, −2.65246676123349163746049736224, 0.859193632346834030970417737182, 2.54629883523220254339790391871, 3.72657715249554248895017139120, 5.74970355939766795666433193192, 5.94664250485621673684292764125, 6.96357375432715058630145399145, 7.928312392832504828264349173179, 9.212706783977088552077286435713, 9.895069172272849188455066475105, 10.64228217469784193387860899549

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