Properties

Label 2-4235-1.1-c1-0-212
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  + 1.27·2-s + 2.68·3-s − 0.367·4-s − 5-s + 3.43·6-s + 7-s − 3.02·8-s + 4.23·9-s − 1.27·10-s − 0.987·12-s − 5.71·13-s + 1.27·14-s − 2.68·15-s − 3.13·16-s + 0.114·17-s + 5.40·18-s − 6.33·19-s + 0.367·20-s + 2.68·21-s − 5.42·23-s − 8.13·24-s + 25-s − 7.30·26-s + 3.31·27-s − 0.367·28-s − 3.52·29-s − 3.43·30-s + ⋯
L(s)  = 1  + 0.903·2-s + 1.55·3-s − 0.183·4-s − 0.447·5-s + 1.40·6-s + 0.377·7-s − 1.06·8-s + 1.41·9-s − 0.404·10-s − 0.285·12-s − 1.58·13-s + 0.341·14-s − 0.694·15-s − 0.782·16-s + 0.0277·17-s + 1.27·18-s − 1.45·19-s + 0.0821·20-s + 0.586·21-s − 1.13·23-s − 1.66·24-s + 0.200·25-s − 1.43·26-s + 0.637·27-s − 0.0694·28-s − 0.655·29-s − 0.627·30-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 \( 1 - 1.27T + 2T^{2} \)
3 \( 1 - 2.68T + 3T^{2} \)
13 \( 1 + 5.71T + 13T^{2} \)
17 \( 1 - 0.114T + 17T^{2} \)
19 \( 1 + 6.33T + 19T^{2} \)
23 \( 1 + 5.42T + 23T^{2} \)
29 \( 1 + 3.52T + 29T^{2} \)
31 \( 1 + 7.76T + 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 - 2.05T + 41T^{2} \)
43 \( 1 + 3.25T + 43T^{2} \)
47 \( 1 + 0.743T + 47T^{2} \)
53 \( 1 - 0.656T + 53T^{2} \)
59 \( 1 + 5.88T + 59T^{2} \)
61 \( 1 + 7.98T + 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 + 0.829T + 71T^{2} \)
73 \( 1 - 1.17T + 73T^{2} \)
79 \( 1 - 0.699T + 79T^{2} \)
83 \( 1 - 8.74T + 83T^{2} \)
89 \( 1 - 8.44T + 89T^{2} \)
97 \( 1 + 7.68T + 97T^{2} \)
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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

−7.950211629376751332093883969586, −7.59281949930163731183308607965, −6.59638670565578232201833032568, −5.63164346319214487849602181155, −4.68058122992059830029739485712, −4.17024520988080786533029659948, −3.52857733101018879474506265543, −2.58090020559562173010432487331, −2.02432368716458933230669943131, 0, 2.02432368716458933230669943131, 2.58090020559562173010432487331, 3.52857733101018879474506265543, 4.17024520988080786533029659948, 4.68058122992059830029739485712, 5.63164346319214487849602181155, 6.59638670565578232201833032568, 7.59281949930163731183308607965, 7.950211629376751332093883969586

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