Properties

Label 2-4235-1.1-c1-0-32
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.637·2-s − 2.56·3-s − 1.59·4-s + 5-s + 1.63·6-s − 7-s + 2.29·8-s + 3.59·9-s − 0.637·10-s + 4.09·12-s − 2.09·13-s + 0.637·14-s − 2.56·15-s + 1.72·16-s + 6.47·17-s − 2.29·18-s + 1.61·19-s − 1.59·20-s + 2.56·21-s + 0.929·23-s − 5.88·24-s + 25-s + 1.33·26-s − 1.52·27-s + 1.59·28-s − 2.41·29-s + 1.63·30-s + ⋯
L(s)  = 1  − 0.451·2-s − 1.48·3-s − 0.796·4-s + 0.447·5-s + 0.668·6-s − 0.377·7-s + 0.810·8-s + 1.19·9-s − 0.201·10-s + 1.18·12-s − 0.579·13-s + 0.170·14-s − 0.662·15-s + 0.431·16-s + 1.57·17-s − 0.540·18-s + 0.369·19-s − 0.356·20-s + 0.560·21-s + 0.193·23-s − 1.20·24-s + 0.200·25-s + 0.261·26-s − 0.293·27-s + 0.301·28-s − 0.447·29-s + 0.299·30-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: \(0\)
Selberg data: \((2,\ 4235,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6203223550\)
\(L(\frac12)\) \(\approx\) \(0.6203223550\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.637T + 2T^{2} \)
3 \( 1 + 2.56T + 3T^{2} \)
13 \( 1 + 2.09T + 13T^{2} \)
17 \( 1 - 6.47T + 17T^{2} \)
19 \( 1 - 1.61T + 19T^{2} \)
23 \( 1 - 0.929T + 23T^{2} \)
29 \( 1 + 2.41T + 29T^{2} \)
31 \( 1 + 0.749T + 31T^{2} \)
37 \( 1 - 1.51T + 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 + 2.78T + 43T^{2} \)
47 \( 1 - 0.403T + 47T^{2} \)
53 \( 1 - 5.79T + 53T^{2} \)
59 \( 1 - 8.93T + 59T^{2} \)
61 \( 1 + 8.13T + 61T^{2} \)
67 \( 1 + 0.789T + 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 + 3.89T + 73T^{2} \)
79 \( 1 + 5.86T + 79T^{2} \)
83 \( 1 + 17.6T + 83T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + 16.6T + 97T^{2} \)
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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

−8.443714715123266493041516733764, −7.51671468180289002557408074245, −7.02757102290435684836327072797, −5.81279724437249517959331384158, −5.66498368436371605570367031470, −4.84595392973163437482729397727, −4.07236619560354025355727151532, −2.94654779255398909215649967950, −1.42369834187236349080823586888, −0.57095863514621023904663267088, 0.57095863514621023904663267088, 1.42369834187236349080823586888, 2.94654779255398909215649967950, 4.07236619560354025355727151532, 4.84595392973163437482729397727, 5.66498368436371605570367031470, 5.81279724437249517959331384158, 7.02757102290435684836327072797, 7.51671468180289002557408074245, 8.443714715123266493041516733764

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