Properties

Label 2-4235-1.1-c1-0-197
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  + 0.300·2-s + 2.68·3-s − 1.90·4-s + 5-s + 0.806·6-s − 7-s − 1.17·8-s + 4.18·9-s + 0.300·10-s − 5.11·12-s − 3.71·13-s − 0.300·14-s + 2.68·15-s + 3.46·16-s − 2.57·17-s + 1.25·18-s − 4.28·19-s − 1.90·20-s − 2.68·21-s − 6.57·23-s − 3.15·24-s + 25-s − 1.11·26-s + 3.18·27-s + 1.90·28-s + 5.39·29-s + 0.806·30-s + ⋯
L(s)  = 1  + 0.212·2-s + 1.54·3-s − 0.954·4-s + 0.447·5-s + 0.329·6-s − 0.377·7-s − 0.415·8-s + 1.39·9-s + 0.0950·10-s − 1.47·12-s − 1.02·13-s − 0.0803·14-s + 0.692·15-s + 0.866·16-s − 0.625·17-s + 0.296·18-s − 0.982·19-s − 0.426·20-s − 0.585·21-s − 1.37·23-s − 0.643·24-s + 0.200·25-s − 0.218·26-s + 0.613·27-s + 0.360·28-s + 1.00·29-s + 0.147·30-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 \( 1 - 0.300T + 2T^{2} \)
3 \( 1 - 2.68T + 3T^{2} \)
13 \( 1 + 3.71T + 13T^{2} \)
17 \( 1 + 2.57T + 17T^{2} \)
19 \( 1 + 4.28T + 19T^{2} \)
23 \( 1 + 6.57T + 23T^{2} \)
29 \( 1 - 5.39T + 29T^{2} \)
31 \( 1 - 0.259T + 31T^{2} \)
37 \( 1 - 8.65T + 37T^{2} \)
41 \( 1 + 5.56T + 41T^{2} \)
43 \( 1 + 7.87T + 43T^{2} \)
47 \( 1 + 7.41T + 47T^{2} \)
53 \( 1 + 3.03T + 53T^{2} \)
59 \( 1 - 9.55T + 59T^{2} \)
61 \( 1 + 4.10T + 61T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 + 5.34T + 71T^{2} \)
73 \( 1 + 3.79T + 73T^{2} \)
79 \( 1 + 16.1T + 79T^{2} \)
83 \( 1 - 6.53T + 83T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 + 3.69T + 97T^{2} \)
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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

−8.368391319320508530450310854454, −7.51661284601716791132604677636, −6.59993424469940915286358978591, −5.82791426288517442065363808662, −4.66580842458492710850646324819, −4.27727981340776338807403344506, −3.28912764668347652667513150028, −2.61400111471674258097005765193, −1.75021700119151168519816621475, 0, 1.75021700119151168519816621475, 2.61400111471674258097005765193, 3.28912764668347652667513150028, 4.27727981340776338807403344506, 4.66580842458492710850646324819, 5.82791426288517442065363808662, 6.59993424469940915286358978591, 7.51661284601716791132604677636, 8.368391319320508530450310854454

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