Properties

Label 2-65e2-1.1-c1-0-40
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  − 1.21·2-s − 2.33·3-s − 0.512·4-s + 2.84·6-s − 3.60·7-s + 3.06·8-s + 2.43·9-s + 5.37·11-s + 1.19·12-s + 4.39·14-s − 2.71·16-s + 1.13·17-s − 2.97·18-s + 2.26·19-s + 8.39·21-s − 6.55·22-s + 3.89·23-s − 7.14·24-s + 1.30·27-s + 1.84·28-s − 0.0247·29-s + 5.46·31-s − 2.81·32-s − 12.5·33-s − 1.38·34-s − 1.24·36-s + 8.70·37-s + ⋯
L(s)  = 1  − 0.862·2-s − 1.34·3-s − 0.256·4-s + 1.16·6-s − 1.36·7-s + 1.08·8-s + 0.813·9-s + 1.61·11-s + 0.344·12-s + 1.17·14-s − 0.678·16-s + 0.274·17-s − 0.701·18-s + 0.520·19-s + 1.83·21-s − 1.39·22-s + 0.811·23-s − 1.45·24-s + 0.251·27-s + 0.348·28-s − 0.00459·29-s + 0.981·31-s − 0.498·32-s − 2.18·33-s − 0.236·34-s − 0.208·36-s + 1.43·37-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

\(L(1)\) \(\approx\) \(0.5592072775\)
\(L(\frac12)\) \(\approx\) \(0.5592072775\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.21T + 2T^{2} \)
3 \( 1 + 2.33T + 3T^{2} \)
7 \( 1 + 3.60T + 7T^{2} \)
11 \( 1 - 5.37T + 11T^{2} \)
17 \( 1 - 1.13T + 17T^{2} \)
19 \( 1 - 2.26T + 19T^{2} \)
23 \( 1 - 3.89T + 23T^{2} \)
29 \( 1 + 0.0247T + 29T^{2} \)
31 \( 1 - 5.46T + 31T^{2} \)
37 \( 1 - 8.70T + 37T^{2} \)
41 \( 1 - 3.73T + 41T^{2} \)
43 \( 1 - 1.13T + 43T^{2} \)
47 \( 1 + 2.58T + 47T^{2} \)
53 \( 1 - 4.43T + 53T^{2} \)
59 \( 1 + 0.171T + 59T^{2} \)
61 \( 1 - 3.36T + 61T^{2} \)
67 \( 1 + 6.39T + 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 - 4.70T + 73T^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + 16.1T + 89T^{2} \)
97 \( 1 + 12.1T + 97T^{2} \)
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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

−8.656249606680855398126132707486, −7.54586206397574884001749398939, −6.84390919895826837906395786438, −6.30322513482039872562830246979, −5.65901171085087409110344300567, −4.65707378579428407107972453239, −3.99021211006077831743406507250, −2.94938307908908578423705918291, −1.26545098273661270542718388480, −0.61417710783734846700004472851, 0.61417710783734846700004472851, 1.26545098273661270542718388480, 2.94938307908908578423705918291, 3.99021211006077831743406507250, 4.65707378579428407107972453239, 5.65901171085087409110344300567, 6.30322513482039872562830246979, 6.84390919895826837906395786438, 7.54586206397574884001749398939, 8.656249606680855398126132707486

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