Properties

Label 2-4235-1.1-c1-0-134
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  − 0.686·2-s + 0.347·3-s − 1.52·4-s − 5-s − 0.238·6-s + 7-s + 2.42·8-s − 2.87·9-s + 0.686·10-s − 0.531·12-s − 6.93·13-s − 0.686·14-s − 0.347·15-s + 1.39·16-s + 1.96·17-s + 1.97·18-s + 6.62·19-s + 1.52·20-s + 0.347·21-s + 0.784·23-s + 0.842·24-s + 25-s + 4.75·26-s − 2.04·27-s − 1.52·28-s + 7.86·29-s + 0.238·30-s + ⋯
L(s)  = 1  − 0.485·2-s + 0.200·3-s − 0.764·4-s − 0.447·5-s − 0.0974·6-s + 0.377·7-s + 0.856·8-s − 0.959·9-s + 0.217·10-s − 0.153·12-s − 1.92·13-s − 0.183·14-s − 0.0897·15-s + 0.348·16-s + 0.475·17-s + 0.465·18-s + 1.51·19-s + 0.341·20-s + 0.0758·21-s + 0.163·23-s + 0.171·24-s + 0.200·25-s + 0.933·26-s − 0.393·27-s − 0.288·28-s + 1.46·29-s + 0.0435·30-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 \( 1 + 0.686T + 2T^{2} \)
3 \( 1 - 0.347T + 3T^{2} \)
13 \( 1 + 6.93T + 13T^{2} \)
17 \( 1 - 1.96T + 17T^{2} \)
19 \( 1 - 6.62T + 19T^{2} \)
23 \( 1 - 0.784T + 23T^{2} \)
29 \( 1 - 7.86T + 29T^{2} \)
31 \( 1 + 0.764T + 31T^{2} \)
37 \( 1 - 3.82T + 37T^{2} \)
41 \( 1 + 7.61T + 41T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 - 6.15T + 53T^{2} \)
59 \( 1 - 7.75T + 59T^{2} \)
61 \( 1 + 5.45T + 61T^{2} \)
67 \( 1 + 5.70T + 67T^{2} \)
71 \( 1 - 4.77T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 + 3.19T + 79T^{2} \)
83 \( 1 + 3.50T + 83T^{2} \)
89 \( 1 - 5.50T + 89T^{2} \)
97 \( 1 - 11.5T + 97T^{2} \)
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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

−7.923697951397775921267327416004, −7.68893734420369985840547197003, −6.84128804830153558804365213563, −5.52722616934504985421745201279, −5.06229893932650488888733180147, −4.36402322415156830125981778052, −3.27034936323214654326712050483, −2.54672216547933649385843692012, −1.12352598462094385700107515612, 0, 1.12352598462094385700107515612, 2.54672216547933649385843692012, 3.27034936323214654326712050483, 4.36402322415156830125981778052, 5.06229893932650488888733180147, 5.52722616934504985421745201279, 6.84128804830153558804365213563, 7.68893734420369985840547197003, 7.923697951397775921267327416004

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