Properties

Label 2-4235-1.1-c1-0-116
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  + 0.414·2-s − 1.41·3-s − 1.82·4-s − 5-s − 0.585·6-s + 7-s − 1.58·8-s − 0.999·9-s − 0.414·10-s + 2.58·12-s − 3.41·13-s + 0.414·14-s + 1.41·15-s + 3·16-s + 0.585·17-s − 0.414·18-s + 1.82·20-s − 1.41·21-s + 8.82·23-s + 2.24·24-s + 25-s − 1.41·26-s + 5.65·27-s − 1.82·28-s + 0.828·29-s + 0.585·30-s − 1.75·31-s + ⋯
L(s)  = 1  + 0.292·2-s − 0.816·3-s − 0.914·4-s − 0.447·5-s − 0.239·6-s + 0.377·7-s − 0.560·8-s − 0.333·9-s − 0.130·10-s + 0.746·12-s − 0.946·13-s + 0.110·14-s + 0.365·15-s + 0.750·16-s + 0.142·17-s − 0.0976·18-s + 0.408·20-s − 0.308·21-s + 1.84·23-s + 0.457·24-s + 0.200·25-s − 0.277·26-s + 1.08·27-s − 0.345·28-s + 0.153·29-s + 0.106·30-s − 0.315·31-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 \( 1 - 0.414T + 2T^{2} \)
3 \( 1 + 1.41T + 3T^{2} \)
13 \( 1 + 3.41T + 13T^{2} \)
17 \( 1 - 0.585T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 8.82T + 23T^{2} \)
29 \( 1 - 0.828T + 29T^{2} \)
31 \( 1 + 1.75T + 31T^{2} \)
37 \( 1 - 1.17T + 37T^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 - 1.41T + 47T^{2} \)
53 \( 1 - 2.82T + 53T^{2} \)
59 \( 1 + 7.89T + 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 - 6.48T + 67T^{2} \)
71 \( 1 - 9.17T + 71T^{2} \)
73 \( 1 - 5.07T + 73T^{2} \)
79 \( 1 + 6.48T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 - 0.828T + 89T^{2} \)
97 \( 1 - 9.31T + 97T^{2} \)
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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

−8.149132046738448833512900903605, −7.19928930490002872794976840561, −6.54432421715936857997355440908, −5.41722959807774290876095240514, −5.15970965587574463859018826269, −4.48086834374553856740265055790, −3.51603340988850335931523430449, −2.65373126249656688844426325396, −1.03955641092666788775615291438, 0, 1.03955641092666788775615291438, 2.65373126249656688844426325396, 3.51603340988850335931523430449, 4.48086834374553856740265055790, 5.15970965587574463859018826269, 5.41722959807774290876095240514, 6.54432421715936857997355440908, 7.19928930490002872794976840561, 8.149132046738448833512900903605

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