Properties

Label 2-4235-1.1-c1-0-207
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  + 1.90·2-s + 0.214·3-s + 1.62·4-s + 5-s + 0.409·6-s − 7-s − 0.710·8-s − 2.95·9-s + 1.90·10-s + 0.349·12-s + 4.56·13-s − 1.90·14-s + 0.214·15-s − 4.60·16-s − 3.25·17-s − 5.62·18-s − 5.02·19-s + 1.62·20-s − 0.214·21-s + 0.292·23-s − 0.152·24-s + 25-s + 8.70·26-s − 1.28·27-s − 1.62·28-s − 1.00·29-s + 0.409·30-s + ⋯
L(s)  = 1  + 1.34·2-s + 0.124·3-s + 0.813·4-s + 0.447·5-s + 0.167·6-s − 0.377·7-s − 0.251·8-s − 0.984·9-s + 0.602·10-s + 0.100·12-s + 1.26·13-s − 0.509·14-s + 0.0555·15-s − 1.15·16-s − 0.788·17-s − 1.32·18-s − 1.15·19-s + 0.363·20-s − 0.0469·21-s + 0.0610·23-s − 0.0311·24-s + 0.200·25-s + 1.70·26-s − 0.246·27-s − 0.307·28-s − 0.186·29-s + 0.0747·30-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 \( 1 - 1.90T + 2T^{2} \)
3 \( 1 - 0.214T + 3T^{2} \)
13 \( 1 - 4.56T + 13T^{2} \)
17 \( 1 + 3.25T + 17T^{2} \)
19 \( 1 + 5.02T + 19T^{2} \)
23 \( 1 - 0.292T + 23T^{2} \)
29 \( 1 + 1.00T + 29T^{2} \)
31 \( 1 + 6.62T + 31T^{2} \)
37 \( 1 + 4.62T + 37T^{2} \)
41 \( 1 + 2.69T + 41T^{2} \)
43 \( 1 + 0.193T + 43T^{2} \)
47 \( 1 + 4.29T + 47T^{2} \)
53 \( 1 - 2.78T + 53T^{2} \)
59 \( 1 + 0.846T + 59T^{2} \)
61 \( 1 + 0.963T + 61T^{2} \)
67 \( 1 - 3.95T + 67T^{2} \)
71 \( 1 - 5.54T + 71T^{2} \)
73 \( 1 + 11.7T + 73T^{2} \)
79 \( 1 - 9.39T + 79T^{2} \)
83 \( 1 + 9.07T + 83T^{2} \)
89 \( 1 - 4.30T + 89T^{2} \)
97 \( 1 - 1.09T + 97T^{2} \)
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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

−8.120917595324659848144030533681, −6.81858007481845255128226852860, −6.38499800838446616774619399483, −5.72260291513597980949499861474, −5.14002367959133247412397782537, −4.12208264666769934794363095592, −3.54385060295725947639879023941, −2.72399455022574741472333559072, −1.85094081854214560250877149282, 0, 1.85094081854214560250877149282, 2.72399455022574741472333559072, 3.54385060295725947639879023941, 4.12208264666769934794363095592, 5.14002367959133247412397782537, 5.72260291513597980949499861474, 6.38499800838446616774619399483, 6.81858007481845255128226852860, 8.120917595324659848144030533681

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