Properties

Label 2-6025-1.1-c1-0-239
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  − 2.44·2-s + 0.992·3-s + 3.99·4-s − 2.42·6-s − 0.744·7-s − 4.88·8-s − 2.01·9-s − 2.60·11-s + 3.96·12-s + 4.18·13-s + 1.82·14-s + 3.96·16-s + 0.484·17-s + 4.93·18-s + 3.30·19-s − 0.738·21-s + 6.37·22-s − 1.44·23-s − 4.84·24-s − 10.2·26-s − 4.97·27-s − 2.97·28-s − 0.129·29-s + 4.43·31-s + 0.0531·32-s − 2.58·33-s − 1.18·34-s + ⋯
L(s)  = 1  − 1.73·2-s + 0.572·3-s + 1.99·4-s − 0.991·6-s − 0.281·7-s − 1.72·8-s − 0.671·9-s − 0.785·11-s + 1.14·12-s + 1.16·13-s + 0.487·14-s + 0.991·16-s + 0.117·17-s + 1.16·18-s + 0.757·19-s − 0.161·21-s + 1.35·22-s − 0.300·23-s − 0.989·24-s − 2.00·26-s − 0.957·27-s − 0.562·28-s − 0.0240·29-s + 0.796·31-s + 0.00939·32-s − 0.449·33-s − 0.203·34-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: $\chi_{6025} (1, \cdot )$
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 \( 1 + 2.44T + 2T^{2} \)
3 \( 1 - 0.992T + 3T^{2} \)
7 \( 1 + 0.744T + 7T^{2} \)
11 \( 1 + 2.60T + 11T^{2} \)
13 \( 1 - 4.18T + 13T^{2} \)
17 \( 1 - 0.484T + 17T^{2} \)
19 \( 1 - 3.30T + 19T^{2} \)
23 \( 1 + 1.44T + 23T^{2} \)
29 \( 1 + 0.129T + 29T^{2} \)
31 \( 1 - 4.43T + 31T^{2} \)
37 \( 1 - 4.67T + 37T^{2} \)
41 \( 1 + 9.21T + 41T^{2} \)
43 \( 1 + 7.65T + 43T^{2} \)
47 \( 1 + 2.28T + 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 + 7.30T + 59T^{2} \)
61 \( 1 + 5.28T + 61T^{2} \)
67 \( 1 - 5.69T + 67T^{2} \)
71 \( 1 - 2.08T + 71T^{2} \)
73 \( 1 - 15.5T + 73T^{2} \)
79 \( 1 + 10.0T + 79T^{2} \)
83 \( 1 + 2.71T + 83T^{2} \)
89 \( 1 + 9.86T + 89T^{2} \)
97 \( 1 - 6.26T + 97T^{2} \)
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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

−8.065750305926808301588600033705, −7.36348198971824695106679518309, −6.52403235325140772432538948981, −5.92386364748736568058089069036, −4.97426110716583212339841897158, −3.59172240729525256023301357905, −2.95993975249728702138874117596, −2.13331510404211575818588154165, −1.14632750826167583598798646459, 0, 1.14632750826167583598798646459, 2.13331510404211575818588154165, 2.95993975249728702138874117596, 3.59172240729525256023301357905, 4.97426110716583212339841897158, 5.92386364748736568058089069036, 6.52403235325140772432538948981, 7.36348198971824695106679518309, 8.065750305926808301588600033705

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