Properties

Label 2-4235-1.1-c1-0-86
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  − 2.74·2-s − 3.04·3-s + 5.51·4-s − 5-s + 8.33·6-s + 7-s − 9.63·8-s + 6.24·9-s + 2.74·10-s − 16.7·12-s − 6.59·13-s − 2.74·14-s + 3.04·15-s + 15.3·16-s + 4.86·17-s − 17.1·18-s − 0.907·19-s − 5.51·20-s − 3.04·21-s + 3.01·23-s + 29.3·24-s + 25-s + 18.0·26-s − 9.87·27-s + 5.51·28-s + 0.914·29-s − 8.33·30-s + ⋯
L(s)  = 1  − 1.93·2-s − 1.75·3-s + 2.75·4-s − 0.447·5-s + 3.40·6-s + 0.377·7-s − 3.40·8-s + 2.08·9-s + 0.866·10-s − 4.84·12-s − 1.82·13-s − 0.732·14-s + 0.785·15-s + 3.84·16-s + 1.17·17-s − 4.03·18-s − 0.208·19-s − 1.23·20-s − 0.663·21-s + 0.628·23-s + 5.98·24-s + 0.200·25-s + 3.54·26-s − 1.90·27-s + 1.04·28-s + 0.169·29-s − 1.52·30-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: Trivial
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 \( 1 + 2.74T + 2T^{2} \)
3 \( 1 + 3.04T + 3T^{2} \)
13 \( 1 + 6.59T + 13T^{2} \)
17 \( 1 - 4.86T + 17T^{2} \)
19 \( 1 + 0.907T + 19T^{2} \)
23 \( 1 - 3.01T + 23T^{2} \)
29 \( 1 - 0.914T + 29T^{2} \)
31 \( 1 + 5.33T + 31T^{2} \)
37 \( 1 + 2.76T + 37T^{2} \)
41 \( 1 - 5.06T + 41T^{2} \)
43 \( 1 + 1.67T + 43T^{2} \)
47 \( 1 - 9.45T + 47T^{2} \)
53 \( 1 + 3.38T + 53T^{2} \)
59 \( 1 + 1.13T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 - 7.50T + 67T^{2} \)
71 \( 1 - 3.74T + 71T^{2} \)
73 \( 1 + 15.8T + 73T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 - 7.76T + 89T^{2} \)
97 \( 1 - 11.3T + 97T^{2} \)
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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

−7.83914706919777983056695749068, −7.28777207240967128813725623540, −7.01322674761328476865988889505, −5.95786968916025561697560646227, −5.40228027001684911541461710758, −4.48914861346375483746686056716, −3.01643043278948826200283704129, −1.84910683627000440871186032933, −0.867792170165655869345317194162, 0, 0.867792170165655869345317194162, 1.84910683627000440871186032933, 3.01643043278948826200283704129, 4.48914861346375483746686056716, 5.40228027001684911541461710758, 5.95786968916025561697560646227, 7.01322674761328476865988889505, 7.28777207240967128813725623540, 7.83914706919777983056695749068

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