Properties

Label 2-6025-1.1-c1-0-352
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  + 2.30·2-s − 0.621·3-s + 3.29·4-s − 1.43·6-s − 0.210·7-s + 2.99·8-s − 2.61·9-s + 2.40·11-s − 2.05·12-s − 0.974·13-s − 0.485·14-s + 0.287·16-s + 0.931·17-s − 6.01·18-s − 4.45·19-s + 0.131·21-s + 5.53·22-s + 3.73·23-s − 1.85·24-s − 2.24·26-s + 3.48·27-s − 0.695·28-s − 6.80·29-s − 10.0·31-s − 5.32·32-s − 1.49·33-s + 2.14·34-s + ⋯
L(s)  = 1  + 1.62·2-s − 0.358·3-s + 1.64·4-s − 0.584·6-s − 0.0797·7-s + 1.05·8-s − 0.871·9-s + 0.724·11-s − 0.591·12-s − 0.270·13-s − 0.129·14-s + 0.0718·16-s + 0.225·17-s − 1.41·18-s − 1.02·19-s + 0.0286·21-s + 1.17·22-s + 0.777·23-s − 0.379·24-s − 0.440·26-s + 0.671·27-s − 0.131·28-s − 1.26·29-s − 1.80·31-s − 0.940·32-s − 0.259·33-s + 0.367·34-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 \( 1 - 2.30T + 2T^{2} \)
3 \( 1 + 0.621T + 3T^{2} \)
7 \( 1 + 0.210T + 7T^{2} \)
11 \( 1 - 2.40T + 11T^{2} \)
13 \( 1 + 0.974T + 13T^{2} \)
17 \( 1 - 0.931T + 17T^{2} \)
19 \( 1 + 4.45T + 19T^{2} \)
23 \( 1 - 3.73T + 23T^{2} \)
29 \( 1 + 6.80T + 29T^{2} \)
31 \( 1 + 10.0T + 31T^{2} \)
37 \( 1 + 6.37T + 37T^{2} \)
41 \( 1 - 2.10T + 41T^{2} \)
43 \( 1 + 4.46T + 43T^{2} \)
47 \( 1 + 3.60T + 47T^{2} \)
53 \( 1 - 9.36T + 53T^{2} \)
59 \( 1 - 0.136T + 59T^{2} \)
61 \( 1 - 14.9T + 61T^{2} \)
67 \( 1 - 1.71T + 67T^{2} \)
71 \( 1 - 8.78T + 71T^{2} \)
73 \( 1 + 11.8T + 73T^{2} \)
79 \( 1 + 2.51T + 79T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 + 0.0938T + 89T^{2} \)
97 \( 1 - 5.72T + 97T^{2} \)
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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

−7.26201888983122865890212104940, −6.80352211307725623102956928705, −6.06859777890692650243125079900, −5.43110144791522234888520117714, −5.00151435910342666033161983582, −3.96095302104661885070773438173, −3.53261364397017495689947038146, −2.59927757341469938194396631734, −1.72152946101527512765930715354, 0, 1.72152946101527512765930715354, 2.59927757341469938194396631734, 3.53261364397017495689947038146, 3.96095302104661885070773438173, 5.00151435910342666033161983582, 5.43110144791522234888520117714, 6.06859777890692650243125079900, 6.80352211307725623102956928705, 7.26201888983122865890212104940

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