Properties

Label 2-6025-1.1-c1-0-38
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  + 1.55·2-s − 2.23·3-s + 0.414·4-s − 3.48·6-s − 1.09·7-s − 2.46·8-s + 2.01·9-s − 0.839·11-s − 0.927·12-s + 3.66·13-s − 1.70·14-s − 4.65·16-s − 4.65·17-s + 3.13·18-s − 5.87·19-s + 2.46·21-s − 1.30·22-s − 4.46·23-s + 5.51·24-s + 5.69·26-s + 2.20·27-s − 0.455·28-s − 5.05·29-s + 6.44·31-s − 2.30·32-s + 1.88·33-s − 7.23·34-s + ⋯
L(s)  = 1  + 1.09·2-s − 1.29·3-s + 0.207·4-s − 1.42·6-s − 0.415·7-s − 0.871·8-s + 0.672·9-s − 0.253·11-s − 0.267·12-s + 1.01·13-s − 0.456·14-s − 1.16·16-s − 1.12·17-s + 0.738·18-s − 1.34·19-s + 0.537·21-s − 0.278·22-s − 0.931·23-s + 1.12·24-s + 1.11·26-s + 0.424·27-s − 0.0860·28-s − 0.938·29-s + 1.15·31-s − 0.408·32-s + 0.327·33-s − 1.24·34-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

\(L(1)\) \(\approx\) \(0.8422130375\)
\(L(\frac12)\) \(\approx\) \(0.8422130375\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.55T + 2T^{2} \)
3 \( 1 + 2.23T + 3T^{2} \)
7 \( 1 + 1.09T + 7T^{2} \)
11 \( 1 + 0.839T + 11T^{2} \)
13 \( 1 - 3.66T + 13T^{2} \)
17 \( 1 + 4.65T + 17T^{2} \)
19 \( 1 + 5.87T + 19T^{2} \)
23 \( 1 + 4.46T + 23T^{2} \)
29 \( 1 + 5.05T + 29T^{2} \)
31 \( 1 - 6.44T + 31T^{2} \)
37 \( 1 - 7.00T + 37T^{2} \)
41 \( 1 + 7.78T + 41T^{2} \)
43 \( 1 + 7.75T + 43T^{2} \)
47 \( 1 - 9.26T + 47T^{2} \)
53 \( 1 + 7.09T + 53T^{2} \)
59 \( 1 - 10.1T + 59T^{2} \)
61 \( 1 + 6.38T + 61T^{2} \)
67 \( 1 + 2.50T + 67T^{2} \)
71 \( 1 + 1.43T + 71T^{2} \)
73 \( 1 - 6.20T + 73T^{2} \)
79 \( 1 + 0.245T + 79T^{2} \)
83 \( 1 - 3.00T + 83T^{2} \)
89 \( 1 + 4.32T + 89T^{2} \)
97 \( 1 - 16.8T + 97T^{2} \)
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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

−8.120571014328731612756356843603, −6.78193787081675355940884376290, −6.35667712612548855580777698486, −5.97990875921096508832437082034, −5.21365170887873072222578704365, −4.47332755753238746560821426600, −3.99267326266836072082814494317, −3.02847758021974065488638306937, −1.97154428168713906177479588562, −0.41582122859605338405388142415, 0.41582122859605338405388142415, 1.97154428168713906177479588562, 3.02847758021974065488638306937, 3.99267326266836072082814494317, 4.47332755753238746560821426600, 5.21365170887873072222578704365, 5.97990875921096508832437082034, 6.35667712612548855580777698486, 6.78193787081675355940884376290, 8.120571014328731612756356843603

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