Properties

Label 2-4235-1.1-c1-0-163
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  − 0.941·2-s + 1.53·3-s − 1.11·4-s − 5-s − 1.44·6-s + 7-s + 2.93·8-s − 0.628·9-s + 0.941·10-s − 1.71·12-s + 1.39·13-s − 0.941·14-s − 1.53·15-s − 0.533·16-s − 2.87·17-s + 0.591·18-s − 1.39·19-s + 1.11·20-s + 1.53·21-s + 6.46·23-s + 4.51·24-s + 25-s − 1.31·26-s − 5.58·27-s − 1.11·28-s − 2.72·29-s + 1.44·30-s + ⋯
L(s)  = 1  − 0.665·2-s + 0.889·3-s − 0.556·4-s − 0.447·5-s − 0.591·6-s + 0.377·7-s + 1.03·8-s − 0.209·9-s + 0.297·10-s − 0.494·12-s + 0.385·13-s − 0.251·14-s − 0.397·15-s − 0.133·16-s − 0.697·17-s + 0.139·18-s − 0.319·19-s + 0.248·20-s + 0.336·21-s + 1.34·23-s + 0.921·24-s + 0.200·25-s − 0.256·26-s − 1.07·27-s − 0.210·28-s − 0.506·29-s + 0.264·30-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 \( 1 + 0.941T + 2T^{2} \)
3 \( 1 - 1.53T + 3T^{2} \)
13 \( 1 - 1.39T + 13T^{2} \)
17 \( 1 + 2.87T + 17T^{2} \)
19 \( 1 + 1.39T + 19T^{2} \)
23 \( 1 - 6.46T + 23T^{2} \)
29 \( 1 + 2.72T + 29T^{2} \)
31 \( 1 + 2.10T + 31T^{2} \)
37 \( 1 + 6.54T + 37T^{2} \)
41 \( 1 + 7.54T + 41T^{2} \)
43 \( 1 - 5.24T + 43T^{2} \)
47 \( 1 - 8.87T + 47T^{2} \)
53 \( 1 - 7.31T + 53T^{2} \)
59 \( 1 + 8.87T + 59T^{2} \)
61 \( 1 + 1.70T + 61T^{2} \)
67 \( 1 - 0.00145T + 67T^{2} \)
71 \( 1 - 6.94T + 71T^{2} \)
73 \( 1 - 0.0957T + 73T^{2} \)
79 \( 1 + 9.24T + 79T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 - 0.516T + 89T^{2} \)
97 \( 1 - 0.474T + 97T^{2} \)
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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

−8.324391645239516917215591696323, −7.48316565019553880015229478666, −7.00024861771781263219377589833, −5.73993423170798917046638867400, −4.92731712964669238710548611981, −4.10398746161757936347970932605, −3.41926448975827386826530331147, −2.37932799084057862861877294386, −1.33389879330898487803248102556, 0, 1.33389879330898487803248102556, 2.37932799084057862861877294386, 3.41926448975827386826530331147, 4.10398746161757936347970932605, 4.92731712964669238710548611981, 5.73993423170798917046638867400, 7.00024861771781263219377589833, 7.48316565019553880015229478666, 8.324391645239516917215591696323

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