Properties

Label 2-65e2-1.1-c1-0-51
Degree $2$
Conductor $4225$
Sign $1$
Analytic cond. $33.7367$
Root an. cond. $5.80833$
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  − 2.54·2-s + 2.15·3-s + 4.48·4-s − 5.48·6-s − 2.93·7-s − 6.31·8-s + 1.63·9-s + 0.635·11-s + 9.64·12-s + 7.48·14-s + 7.11·16-s + 1.22·17-s − 4.16·18-s − 1.36·19-s − 6.32·21-s − 1.61·22-s − 2.15·23-s − 13.5·24-s − 2.93·27-s − 13.1·28-s + 3·29-s + 8.96·31-s − 5.48·32-s + 1.36·33-s − 3.11·34-s + 7.32·36-s + 1.22·37-s + ⋯
L(s)  = 1  − 1.80·2-s + 1.24·3-s + 2.24·4-s − 2.23·6-s − 1.11·7-s − 2.23·8-s + 0.545·9-s + 0.191·11-s + 2.78·12-s + 1.99·14-s + 1.77·16-s + 0.296·17-s − 0.981·18-s − 0.313·19-s − 1.38·21-s − 0.344·22-s − 0.448·23-s − 2.77·24-s − 0.565·27-s − 2.48·28-s + 0.557·29-s + 1.60·31-s − 0.969·32-s + 0.238·33-s − 0.534·34-s + 1.22·36-s + 0.201·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(33.7367\)
Root analytic conductor: \(5.80833\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4225,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9865362460\)
\(L(\frac12)\) \(\approx\) \(0.9865362460\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + 2.54T + 2T^{2} \)
3 \( 1 - 2.15T + 3T^{2} \)
7 \( 1 + 2.93T + 7T^{2} \)
11 \( 1 - 0.635T + 11T^{2} \)
17 \( 1 - 1.22T + 17T^{2} \)
19 \( 1 + 1.36T + 19T^{2} \)
23 \( 1 + 2.15T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 - 8.96T + 31T^{2} \)
37 \( 1 - 1.22T + 37T^{2} \)
41 \( 1 - 9.96T + 41T^{2} \)
43 \( 1 - 1.36T + 43T^{2} \)
47 \( 1 + 6.16T + 47T^{2} \)
53 \( 1 + 0.642T + 53T^{2} \)
59 \( 1 + 7.59T + 59T^{2} \)
61 \( 1 + 2.27T + 61T^{2} \)
67 \( 1 - 8.03T + 67T^{2} \)
71 \( 1 + 2.63T + 71T^{2} \)
73 \( 1 + 10.3T + 73T^{2} \)
79 \( 1 - 1.03T + 79T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 - 12.5T + 89T^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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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

−8.381890467899477972915173683994, −8.014967820609355596927211343036, −7.29731978240768510851812754430, −6.49190948404918613439438672048, −5.98964917599266089606954172160, −4.37800626209456910675083352982, −3.25018054997023861453610621300, −2.75110357565396219046321767923, −1.88622196251371329653174069856, −0.67617993828912182715233039440, 0.67617993828912182715233039440, 1.88622196251371329653174069856, 2.75110357565396219046321767923, 3.25018054997023861453610621300, 4.37800626209456910675083352982, 5.98964917599266089606954172160, 6.49190948404918613439438672048, 7.29731978240768510851812754430, 8.014967820609355596927211343036, 8.381890467899477972915173683994

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