Properties

Label 2-6025-1.1-c1-0-2
Degree $2$
Conductor $6025$
Sign $1$
Analytic cond. $48.1098$
Root an. cond. $6.93612$
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  − 0.337·2-s − 2.44·3-s − 1.88·4-s + 0.824·6-s − 2.26·7-s + 1.31·8-s + 2.98·9-s − 4.20·11-s + 4.61·12-s − 1.32·13-s + 0.764·14-s + 3.33·16-s + 1.12·17-s − 1.00·18-s − 1.53·19-s + 5.54·21-s + 1.41·22-s + 5.10·23-s − 3.20·24-s + 0.447·26-s + 0.0377·27-s + 4.27·28-s − 5.78·29-s − 4.45·31-s − 3.74·32-s + 10.2·33-s − 0.377·34-s + ⋯
L(s)  = 1  − 0.238·2-s − 1.41·3-s − 0.943·4-s + 0.336·6-s − 0.857·7-s + 0.463·8-s + 0.994·9-s − 1.26·11-s + 1.33·12-s − 0.367·13-s + 0.204·14-s + 0.832·16-s + 0.271·17-s − 0.237·18-s − 0.351·19-s + 1.21·21-s + 0.302·22-s + 1.06·23-s − 0.654·24-s + 0.0877·26-s + 0.00726·27-s + 0.808·28-s − 1.07·29-s − 0.800·31-s − 0.661·32-s + 1.78·33-s − 0.0648·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $1$
Analytic conductor: \(48.1098\)
Root analytic conductor: \(6.93612\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.01225889564\)
\(L(\frac12)\) \(\approx\) \(0.01225889564\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 - T \)
good2 \( 1 + 0.337T + 2T^{2} \)
3 \( 1 + 2.44T + 3T^{2} \)
7 \( 1 + 2.26T + 7T^{2} \)
11 \( 1 + 4.20T + 11T^{2} \)
13 \( 1 + 1.32T + 13T^{2} \)
17 \( 1 - 1.12T + 17T^{2} \)
19 \( 1 + 1.53T + 19T^{2} \)
23 \( 1 - 5.10T + 23T^{2} \)
29 \( 1 + 5.78T + 29T^{2} \)
31 \( 1 + 4.45T + 31T^{2} \)
37 \( 1 + 10.8T + 37T^{2} \)
41 \( 1 + 3.09T + 41T^{2} \)
43 \( 1 - 8.63T + 43T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 + 0.172T + 53T^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 - 7.54T + 67T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 + 1.82T + 73T^{2} \)
79 \( 1 + 4.65T + 79T^{2} \)
83 \( 1 + 6.24T + 83T^{2} \)
89 \( 1 + 3.91T + 89T^{2} \)
97 \( 1 + 10.7T + 97T^{2} \)
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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

−8.007370762188650108485706929153, −7.31023545791518536456948949963, −6.61262527850345970869131260906, −5.75379676107662970521456800668, −5.21961102871300978801058631904, −4.79608315346049142267142599136, −3.73853592125752858427876313426, −2.90648075656017206350690733364, −1.46752874052682508912412514373, −0.06729908877779492902533602279, 0.06729908877779492902533602279, 1.46752874052682508912412514373, 2.90648075656017206350690733364, 3.73853592125752858427876313426, 4.79608315346049142267142599136, 5.21961102871300978801058631904, 5.75379676107662970521456800668, 6.61262527850345970869131260906, 7.31023545791518536456948949963, 8.007370762188650108485706929153

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