Properties

Label 2-65e2-1.1-c1-0-14
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  − 0.271·2-s + 0.319·3-s − 1.92·4-s − 0.0867·6-s − 3.38·7-s + 1.06·8-s − 2.89·9-s + 1.75·11-s − 0.615·12-s + 0.917·14-s + 3.56·16-s − 1.95·17-s + 0.786·18-s − 7.13·19-s − 1.08·21-s − 0.475·22-s − 7.61·23-s + 0.340·24-s − 1.88·27-s + 6.51·28-s + 3.98·29-s + 4.86·31-s − 3.09·32-s + 0.559·33-s + 0.530·34-s + 5.58·36-s − 10.5·37-s + ⋯
L(s)  = 1  − 0.191·2-s + 0.184·3-s − 0.963·4-s − 0.0354·6-s − 1.27·7-s + 0.376·8-s − 0.965·9-s + 0.528·11-s − 0.177·12-s + 0.245·14-s + 0.890·16-s − 0.473·17-s + 0.185·18-s − 1.63·19-s − 0.235·21-s − 0.101·22-s − 1.58·23-s + 0.0695·24-s − 0.362·27-s + 1.23·28-s + 0.739·29-s + 0.874·31-s − 0.547·32-s + 0.0974·33-s + 0.0909·34-s + 0.930·36-s − 1.72·37-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4225,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4567371224\)
\(L(\frac12)\) \(\approx\) \(0.4567371224\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.271T + 2T^{2} \)
3 \( 1 - 0.319T + 3T^{2} \)
7 \( 1 + 3.38T + 7T^{2} \)
11 \( 1 - 1.75T + 11T^{2} \)
17 \( 1 + 1.95T + 17T^{2} \)
19 \( 1 + 7.13T + 19T^{2} \)
23 \( 1 + 7.61T + 23T^{2} \)
29 \( 1 - 3.98T + 29T^{2} \)
31 \( 1 - 4.86T + 31T^{2} \)
37 \( 1 + 10.5T + 37T^{2} \)
41 \( 1 - 0.911T + 41T^{2} \)
43 \( 1 + 4.58T + 43T^{2} \)
47 \( 1 - 8.58T + 47T^{2} \)
53 \( 1 + 11.9T + 53T^{2} \)
59 \( 1 - 3.82T + 59T^{2} \)
61 \( 1 - 7.98T + 61T^{2} \)
67 \( 1 + 0.472T + 67T^{2} \)
71 \( 1 + 5.50T + 71T^{2} \)
73 \( 1 + 2.93T + 73T^{2} \)
79 \( 1 - 4.09T + 79T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + 3.85T + 89T^{2} \)
97 \( 1 - 6.95T + 97T^{2} \)
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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

−8.535581146882649443896242173629, −7.985648263483693282076992025888, −6.73515667479236811516395213133, −6.29205728491981516284882061985, −5.52687365358230507614024275048, −4.46863513839017758333313954818, −3.83947116363504224524880063012, −3.05346768738306111773972340242, −1.99494622340803464077276864585, −0.37305578049020378778633239667, 0.37305578049020378778633239667, 1.99494622340803464077276864585, 3.05346768738306111773972340242, 3.83947116363504224524880063012, 4.46863513839017758333313954818, 5.52687365358230507614024275048, 6.29205728491981516284882061985, 6.73515667479236811516395213133, 7.985648263483693282076992025888, 8.535581146882649443896242173629

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