Properties

Label 2-6025-1.1-c1-0-372
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.31·2-s + 1.11·3-s + 3.35·4-s + 2.59·6-s − 1.28·7-s + 3.13·8-s − 1.74·9-s − 1.10·11-s + 3.75·12-s − 3.93·13-s − 2.97·14-s + 0.553·16-s − 2.17·17-s − 4.04·18-s − 3.54·19-s − 1.43·21-s − 2.54·22-s + 5.81·23-s + 3.51·24-s − 9.11·26-s − 5.31·27-s − 4.30·28-s − 4.44·29-s − 0.859·31-s − 4.99·32-s − 1.23·33-s − 5.03·34-s + ⋯
L(s)  = 1  + 1.63·2-s + 0.646·3-s + 1.67·4-s + 1.05·6-s − 0.485·7-s + 1.11·8-s − 0.582·9-s − 0.331·11-s + 1.08·12-s − 1.09·13-s − 0.793·14-s + 0.138·16-s − 0.527·17-s − 0.952·18-s − 0.812·19-s − 0.313·21-s − 0.542·22-s + 1.21·23-s + 0.717·24-s − 1.78·26-s − 1.02·27-s − 0.814·28-s − 0.825·29-s − 0.154·31-s − 0.883·32-s − 0.214·33-s − 0.863·34-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: \(1\)
Selberg data: \((2,\ 6025,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.31T + 2T^{2} \)
3 \( 1 - 1.11T + 3T^{2} \)
7 \( 1 + 1.28T + 7T^{2} \)
11 \( 1 + 1.10T + 11T^{2} \)
13 \( 1 + 3.93T + 13T^{2} \)
17 \( 1 + 2.17T + 17T^{2} \)
19 \( 1 + 3.54T + 19T^{2} \)
23 \( 1 - 5.81T + 23T^{2} \)
29 \( 1 + 4.44T + 29T^{2} \)
31 \( 1 + 0.859T + 31T^{2} \)
37 \( 1 - 3.70T + 37T^{2} \)
41 \( 1 + 3.34T + 41T^{2} \)
43 \( 1 - 2.52T + 43T^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 - 9.19T + 59T^{2} \)
61 \( 1 + 9.71T + 61T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 - 6.93T + 73T^{2} \)
79 \( 1 - 1.95T + 79T^{2} \)
83 \( 1 - 17.1T + 83T^{2} \)
89 \( 1 - 13.6T + 89T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−7.61819261004358774268333430395, −6.62265048975807369115653849726, −6.38357514785423701942727574002, −5.17794001843923857027542982223, −5.01466236419618735628722944120, −3.94506810353449244119400006042, −3.30124853658517274906737931468, −2.61971169472287547996995608861, −2.03734120816558673127556175512, 0, 2.03734120816558673127556175512, 2.61971169472287547996995608861, 3.30124853658517274906737931468, 3.94506810353449244119400006042, 5.01466236419618735628722944120, 5.17794001843923857027542982223, 6.38357514785423701942727574002, 6.62265048975807369115653849726, 7.61819261004358774268333430395

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