Properties

Label 2-4235-1.1-c1-0-119
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  − 1.21·2-s − 0.688·3-s − 0.525·4-s − 5-s + 0.836·6-s + 7-s + 3.06·8-s − 2.52·9-s + 1.21·10-s + 0.361·12-s + 3.73·13-s − 1.21·14-s + 0.688·15-s − 2.67·16-s − 0.0666·17-s + 3.06·18-s − 6.42·19-s + 0.525·20-s − 0.688·21-s − 1.09·23-s − 2.11·24-s + 25-s − 4.54·26-s + 3.80·27-s − 0.525·28-s + 7.80·29-s − 0.836·30-s + ⋯
L(s)  = 1  − 0.858·2-s − 0.397·3-s − 0.262·4-s − 0.447·5-s + 0.341·6-s + 0.377·7-s + 1.08·8-s − 0.841·9-s + 0.384·10-s + 0.104·12-s + 1.03·13-s − 0.324·14-s + 0.177·15-s − 0.668·16-s − 0.0161·17-s + 0.722·18-s − 1.47·19-s + 0.117·20-s − 0.150·21-s − 0.228·23-s − 0.431·24-s + 0.200·25-s − 0.890·26-s + 0.732·27-s − 0.0992·28-s + 1.44·29-s − 0.152·30-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 \( 1 + 1.21T + 2T^{2} \)
3 \( 1 + 0.688T + 3T^{2} \)
13 \( 1 - 3.73T + 13T^{2} \)
17 \( 1 + 0.0666T + 17T^{2} \)
19 \( 1 + 6.42T + 19T^{2} \)
23 \( 1 + 1.09T + 23T^{2} \)
29 \( 1 - 7.80T + 29T^{2} \)
31 \( 1 + 5.59T + 31T^{2} \)
37 \( 1 - 1.33T + 37T^{2} \)
41 \( 1 + 6.64T + 41T^{2} \)
43 \( 1 - 11.7T + 43T^{2} \)
47 \( 1 + 2.26T + 47T^{2} \)
53 \( 1 + 1.71T + 53T^{2} \)
59 \( 1 - 2.54T + 59T^{2} \)
61 \( 1 + 14.4T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 + 12.5T + 71T^{2} \)
73 \( 1 + 1.17T + 73T^{2} \)
79 \( 1 - 8.51T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 + 13.4T + 97T^{2} \)
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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

−8.225598530648414696092947857332, −7.58865313164770722728634366841, −6.57099044643292256822477397115, −5.95727993395610976695065466334, −4.96865993925329474311816956920, −4.32280712671900465423079312014, −3.45966285607318574460234225772, −2.20105613770269576807152234021, −1.05285008971948718870510312235, 0, 1.05285008971948718870510312235, 2.20105613770269576807152234021, 3.45966285607318574460234225772, 4.32280712671900465423079312014, 4.96865993925329474311816956920, 5.95727993395610976695065466334, 6.57099044643292256822477397115, 7.58865313164770722728634366841, 8.225598530648414696092947857332

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