Properties

Label 2-4235-1.1-c1-0-178
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  − 2.43·2-s + 2.36·3-s + 3.92·4-s + 5-s − 5.75·6-s − 7-s − 4.68·8-s + 2.59·9-s − 2.43·10-s + 9.28·12-s − 1.34·13-s + 2.43·14-s + 2.36·15-s + 3.56·16-s − 1.53·17-s − 6.31·18-s + 1.50·19-s + 3.92·20-s − 2.36·21-s + 2.62·23-s − 11.0·24-s + 25-s + 3.26·26-s − 0.958·27-s − 3.92·28-s − 7.31·29-s − 5.75·30-s + ⋯
L(s)  = 1  − 1.72·2-s + 1.36·3-s + 1.96·4-s + 0.447·5-s − 2.35·6-s − 0.377·7-s − 1.65·8-s + 0.864·9-s − 0.769·10-s + 2.68·12-s − 0.371·13-s + 0.650·14-s + 0.610·15-s + 0.890·16-s − 0.371·17-s − 1.48·18-s + 0.344·19-s + 0.877·20-s − 0.516·21-s + 0.548·23-s − 2.26·24-s + 0.200·25-s + 0.639·26-s − 0.184·27-s − 0.741·28-s − 1.35·29-s − 1.05·30-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 \( 1 + 2.43T + 2T^{2} \)
3 \( 1 - 2.36T + 3T^{2} \)
13 \( 1 + 1.34T + 13T^{2} \)
17 \( 1 + 1.53T + 17T^{2} \)
19 \( 1 - 1.50T + 19T^{2} \)
23 \( 1 - 2.62T + 23T^{2} \)
29 \( 1 + 7.31T + 29T^{2} \)
31 \( 1 + 7.31T + 31T^{2} \)
37 \( 1 + 11.6T + 37T^{2} \)
41 \( 1 + 4.01T + 41T^{2} \)
43 \( 1 - 12.8T + 43T^{2} \)
47 \( 1 + 6.51T + 47T^{2} \)
53 \( 1 + 0.974T + 53T^{2} \)
59 \( 1 + 0.145T + 59T^{2} \)
61 \( 1 + 4.83T + 61T^{2} \)
67 \( 1 + 6.74T + 67T^{2} \)
71 \( 1 - 4.55T + 71T^{2} \)
73 \( 1 - 5.13T + 73T^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 - 3.80T + 83T^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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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

−8.273217166449953938804627257170, −7.35401244485502855990033885333, −7.22296809738086561006719783875, −6.17391677770821248876424645151, −5.18181753617336024621698011764, −3.79887494849885598640095135593, −2.99503999804093788544092266048, −2.17929196518869179665182134975, −1.53625436142024455365372786184, 0, 1.53625436142024455365372786184, 2.17929196518869179665182134975, 2.99503999804093788544092266048, 3.79887494849885598640095135593, 5.18181753617336024621698011764, 6.17391677770821248876424645151, 7.22296809738086561006719783875, 7.35401244485502855990033885333, 8.273217166449953938804627257170

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