Properties

Label 2-6025-1.1-c1-0-310
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-s + 1.61·3-s − 4-s − 1.61·6-s + 1.23·7-s + 3·8-s − 0.381·9-s + 3.61·11-s − 1.61·12-s + 0.381·13-s − 1.23·14-s − 16-s + 3.85·17-s + 0.381·18-s + 0.618·19-s + 2.00·21-s − 3.61·22-s − 8.47·23-s + 4.85·24-s − 0.381·26-s − 5.47·27-s − 1.23·28-s − 7.09·29-s − 1.52·31-s − 5·32-s + 5.85·33-s − 3.85·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.934·3-s − 0.5·4-s − 0.660·6-s + 0.467·7-s + 1.06·8-s − 0.127·9-s + 1.09·11-s − 0.467·12-s + 0.105·13-s − 0.330·14-s − 0.250·16-s + 0.934·17-s + 0.0900·18-s + 0.141·19-s + 0.436·21-s − 0.771·22-s − 1.76·23-s + 0.990·24-s − 0.0749·26-s − 1.05·27-s − 0.233·28-s − 1.31·29-s − 0.274·31-s − 0.883·32-s + 1.01·33-s − 0.660·34-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 \( 1 + T + 2T^{2} \)
3 \( 1 - 1.61T + 3T^{2} \)
7 \( 1 - 1.23T + 7T^{2} \)
11 \( 1 - 3.61T + 11T^{2} \)
13 \( 1 - 0.381T + 13T^{2} \)
17 \( 1 - 3.85T + 17T^{2} \)
19 \( 1 - 0.618T + 19T^{2} \)
23 \( 1 + 8.47T + 23T^{2} \)
29 \( 1 + 7.09T + 29T^{2} \)
31 \( 1 + 1.52T + 31T^{2} \)
37 \( 1 + 4.47T + 37T^{2} \)
41 \( 1 + 4.09T + 41T^{2} \)
43 \( 1 + 6.47T + 43T^{2} \)
47 \( 1 + 7.85T + 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + 2.76T + 59T^{2} \)
61 \( 1 - 4.85T + 61T^{2} \)
67 \( 1 + 12.7T + 67T^{2} \)
71 \( 1 + 14.7T + 71T^{2} \)
73 \( 1 - 1.09T + 73T^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 - 16.4T + 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−7.893288311115908960591392915004, −7.47035598278148838144068080098, −6.40295140482344952956253854917, −5.56067739061899013051516328548, −4.75054714234430280442058207794, −3.74254638290378031701523113178, −3.46493232716067488042819771296, −1.99330548440619093424174483893, −1.46516315221642415396202937985, 0, 1.46516315221642415396202937985, 1.99330548440619093424174483893, 3.46493232716067488042819771296, 3.74254638290378031701523113178, 4.75054714234430280442058207794, 5.56067739061899013051516328548, 6.40295140482344952956253854917, 7.47035598278148838144068080098, 7.893288311115908960591392915004

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