Properties

Label 2-4235-1.1-c1-0-70
Degree $2$
Conductor $4235$
Sign $1$
Analytic cond. $33.8166$
Root an. cond. $5.81520$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.940·2-s + 0.854·3-s − 1.11·4-s + 5-s + 0.804·6-s + 7-s − 2.93·8-s − 2.26·9-s + 0.940·10-s − 0.953·12-s − 5.89·13-s + 0.940·14-s + 0.854·15-s − 0.525·16-s + 6.46·17-s − 2.13·18-s + 3.18·19-s − 1.11·20-s + 0.854·21-s + 0.468·23-s − 2.50·24-s + 25-s − 5.54·26-s − 4.50·27-s − 1.11·28-s − 0.0101·29-s + 0.804·30-s + ⋯
L(s)  = 1  + 0.665·2-s + 0.493·3-s − 0.557·4-s + 0.447·5-s + 0.328·6-s + 0.377·7-s − 1.03·8-s − 0.756·9-s + 0.297·10-s − 0.275·12-s − 1.63·13-s + 0.251·14-s + 0.220·15-s − 0.131·16-s + 1.56·17-s − 0.503·18-s + 0.729·19-s − 0.249·20-s + 0.186·21-s + 0.0977·23-s − 0.511·24-s + 0.200·25-s − 1.08·26-s − 0.866·27-s − 0.210·28-s − 0.00187·29-s + 0.146·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(33.8166\)
Root analytic conductor: \(5.81520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4235,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.584465815\)
\(L(\frac12)\) \(\approx\) \(2.584465815\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 \)
good2 \( 1 - 0.940T + 2T^{2} \)
3 \( 1 - 0.854T + 3T^{2} \)
13 \( 1 + 5.89T + 13T^{2} \)
17 \( 1 - 6.46T + 17T^{2} \)
19 \( 1 - 3.18T + 19T^{2} \)
23 \( 1 - 0.468T + 23T^{2} \)
29 \( 1 + 0.0101T + 29T^{2} \)
31 \( 1 - 3.76T + 31T^{2} \)
37 \( 1 - 9.86T + 37T^{2} \)
41 \( 1 - 9.10T + 41T^{2} \)
43 \( 1 + 12.2T + 43T^{2} \)
47 \( 1 + 8.76T + 47T^{2} \)
53 \( 1 - 9.30T + 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 - 3.75T + 61T^{2} \)
67 \( 1 - 6.56T + 67T^{2} \)
71 \( 1 - 6.07T + 71T^{2} \)
73 \( 1 + 1.27T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 - 7.71T + 97T^{2} \)
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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

−8.182305194353586589135025410896, −7.936739578460875489064498749910, −6.90849237158207906741935737017, −5.88533968336918059285226373203, −5.29216923242456313701282579375, −4.82962641124930390227967181389, −3.76297451052891659653032774393, −2.97398355872775235424762444975, −2.33524234241780629532574998949, −0.800463899913976810419887841736, 0.800463899913976810419887841736, 2.33524234241780629532574998949, 2.97398355872775235424762444975, 3.76297451052891659653032774393, 4.82962641124930390227967181389, 5.29216923242456313701282579375, 5.88533968336918059285226373203, 6.90849237158207906741935737017, 7.936739578460875489064498749910, 8.182305194353586589135025410896

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