Properties

Label 2-4235-1.1-c1-0-196
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  + 2.67·2-s − 2.48·3-s + 5.15·4-s − 5-s − 6.63·6-s + 7-s + 8.44·8-s + 3.15·9-s − 2.67·10-s − 12.7·12-s − 5.83·13-s + 2.67·14-s + 2.48·15-s + 12.2·16-s − 5.44·17-s + 8.44·18-s + 1.35·19-s − 5.15·20-s − 2.48·21-s − 3.19·23-s − 20.9·24-s + 25-s − 15.5·26-s − 0.387·27-s + 5.15·28-s + 3.61·29-s + 6.63·30-s + ⋯
L(s)  = 1  + 1.89·2-s − 1.43·3-s + 2.57·4-s − 0.447·5-s − 2.70·6-s + 0.377·7-s + 2.98·8-s + 1.05·9-s − 0.845·10-s − 3.69·12-s − 1.61·13-s + 0.714·14-s + 0.640·15-s + 3.06·16-s − 1.32·17-s + 1.99·18-s + 0.309·19-s − 1.15·20-s − 0.541·21-s − 0.665·23-s − 4.27·24-s + 0.200·25-s − 3.05·26-s − 0.0746·27-s + 0.974·28-s + 0.670·29-s + 1.21·30-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 \( 1 - 2.67T + 2T^{2} \)
3 \( 1 + 2.48T + 3T^{2} \)
13 \( 1 + 5.83T + 13T^{2} \)
17 \( 1 + 5.44T + 17T^{2} \)
19 \( 1 - 1.35T + 19T^{2} \)
23 \( 1 + 3.19T + 23T^{2} \)
29 \( 1 - 3.61T + 29T^{2} \)
31 \( 1 + 5.28T + 31T^{2} \)
37 \( 1 + 8.54T + 37T^{2} \)
41 \( 1 - 5.02T + 41T^{2} \)
43 \( 1 + 5.89T + 43T^{2} \)
47 \( 1 + 11.8T + 47T^{2} \)
53 \( 1 + 0.231T + 53T^{2} \)
59 \( 1 - 13.5T + 59T^{2} \)
61 \( 1 - 1.41T + 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 + 15.5T + 71T^{2} \)
73 \( 1 - 11.3T + 73T^{2} \)
79 \( 1 + 1.96T + 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 - 7.22T + 89T^{2} \)
97 \( 1 + 0.836T + 97T^{2} \)
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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

−7.41521569550597947372831559612, −7.00116642646110917256463750055, −6.35011428390453590773251296220, −5.57469812107072089090735828244, −4.88602809142225917595625630127, −4.65297363146386151934892208912, −3.74058465021985711254123288528, −2.66160670152634085717806283879, −1.74103237889578837106585752051, 0, 1.74103237889578837106585752051, 2.66160670152634085717806283879, 3.74058465021985711254123288528, 4.65297363146386151934892208912, 4.88602809142225917595625630127, 5.57469812107072089090735828244, 6.35011428390453590773251296220, 7.00116642646110917256463750055, 7.41521569550597947372831559612

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