Properties

Label 2-4235-1.1-c1-0-115
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.65·2-s − 1.14·3-s + 0.734·4-s − 5-s + 1.88·6-s + 7-s + 2.09·8-s − 1.69·9-s + 1.65·10-s − 0.838·12-s + 5.60·13-s − 1.65·14-s + 1.14·15-s − 4.92·16-s − 4.57·17-s + 2.80·18-s + 1.88·19-s − 0.734·20-s − 1.14·21-s + 2.08·23-s − 2.38·24-s + 25-s − 9.26·26-s + 5.36·27-s + 0.734·28-s − 5.51·29-s − 1.88·30-s + ⋯
L(s)  = 1  − 1.16·2-s − 0.659·3-s + 0.367·4-s − 0.447·5-s + 0.770·6-s + 0.377·7-s + 0.739·8-s − 0.565·9-s + 0.522·10-s − 0.242·12-s + 1.55·13-s − 0.441·14-s + 0.294·15-s − 1.23·16-s − 1.11·17-s + 0.661·18-s + 0.431·19-s − 0.164·20-s − 0.249·21-s + 0.433·23-s − 0.487·24-s + 0.200·25-s − 1.81·26-s + 1.03·27-s + 0.138·28-s − 1.02·29-s − 0.344·30-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: $\chi_{4235} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4235,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.65T + 2T^{2} \)
3 \( 1 + 1.14T + 3T^{2} \)
13 \( 1 - 5.60T + 13T^{2} \)
17 \( 1 + 4.57T + 17T^{2} \)
19 \( 1 - 1.88T + 19T^{2} \)
23 \( 1 - 2.08T + 23T^{2} \)
29 \( 1 + 5.51T + 29T^{2} \)
31 \( 1 + 5.46T + 31T^{2} \)
37 \( 1 + 9.34T + 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 + 8.43T + 43T^{2} \)
47 \( 1 - 1.60T + 47T^{2} \)
53 \( 1 - 6.21T + 53T^{2} \)
59 \( 1 - 13.5T + 59T^{2} \)
61 \( 1 + 11.1T + 61T^{2} \)
67 \( 1 + 5.60T + 67T^{2} \)
71 \( 1 - 0.696T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 + 8.47T + 79T^{2} \)
83 \( 1 + 9.50T + 83T^{2} \)
89 \( 1 - 1.49T + 89T^{2} \)
97 \( 1 - 5.93T + 97T^{2} \)
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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

−8.266738882205082784186440670241, −7.39414972554395377339443733384, −6.79665000311362658640308343577, −5.87090610975042762074685331209, −5.16513244309112556069637408000, −4.25206626180890219927236749433, −3.42583463877008724542717314681, −2.04610345758136170996818903819, −1.03918884422847804902374179028, 0, 1.03918884422847804902374179028, 2.04610345758136170996818903819, 3.42583463877008724542717314681, 4.25206626180890219927236749433, 5.16513244309112556069637408000, 5.87090610975042762074685331209, 6.79665000311362658640308343577, 7.39414972554395377339443733384, 8.266738882205082784186440670241

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