Properties

Label 2-4235-1.1-c1-0-98
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  − 2.23·2-s − 1.61·3-s + 3.00·4-s − 5-s + 3.61·6-s + 7-s − 2.23·8-s − 0.381·9-s + 2.23·10-s − 4.85·12-s − 2.38·13-s − 2.23·14-s + 1.61·15-s − 0.999·16-s − 3.38·17-s + 0.854·18-s − 1.23·19-s − 3.00·20-s − 1.61·21-s + 4.47·23-s + 3.61·24-s + 25-s + 5.32·26-s + 5.47·27-s + 3.00·28-s + 5.85·29-s − 3.61·30-s + ⋯
L(s)  = 1  − 1.58·2-s − 0.934·3-s + 1.50·4-s − 0.447·5-s + 1.47·6-s + 0.377·7-s − 0.790·8-s − 0.127·9-s + 0.707·10-s − 1.40·12-s − 0.660·13-s − 0.597·14-s + 0.417·15-s − 0.249·16-s − 0.820·17-s + 0.201·18-s − 0.283·19-s − 0.670·20-s − 0.353·21-s + 0.932·23-s + 0.738·24-s + 0.200·25-s + 1.04·26-s + 1.05·27-s + 0.566·28-s + 1.08·29-s − 0.660·30-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 \( 1 + 2.23T + 2T^{2} \)
3 \( 1 + 1.61T + 3T^{2} \)
13 \( 1 + 2.38T + 13T^{2} \)
17 \( 1 + 3.38T + 17T^{2} \)
19 \( 1 + 1.23T + 19T^{2} \)
23 \( 1 - 4.47T + 23T^{2} \)
29 \( 1 - 5.85T + 29T^{2} \)
31 \( 1 + 1.23T + 31T^{2} \)
37 \( 1 + 2.76T + 37T^{2} \)
41 \( 1 - 6.47T + 41T^{2} \)
43 \( 1 - 0.763T + 43T^{2} \)
47 \( 1 + 7.85T + 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + 4.76T + 59T^{2} \)
61 \( 1 - 9.23T + 61T^{2} \)
67 \( 1 + 14.1T + 67T^{2} \)
71 \( 1 - 0.673T + 71T^{2} \)
73 \( 1 + 8.61T + 73T^{2} \)
79 \( 1 - 13.7T + 79T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 - 0.326T + 97T^{2} \)
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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

−8.092582659500586959166230802670, −7.46465613488651781385076081105, −6.71859290254297040118163411549, −6.17792246521172978695981051175, −4.97590174638285796663307971688, −4.55749428408365161686413029873, −3.07106680839664990760492274157, −2.08133672554044300140068799309, −0.920308906870214704398971553972, 0, 0.920308906870214704398971553972, 2.08133672554044300140068799309, 3.07106680839664990760492274157, 4.55749428408365161686413029873, 4.97590174638285796663307971688, 6.17792246521172978695981051175, 6.71859290254297040118163411549, 7.46465613488651781385076081105, 8.092582659500586959166230802670

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