Properties

Label 2-6025-1.1-c1-0-205
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  − 0.810·2-s + 2.88·3-s − 1.34·4-s − 2.33·6-s + 3.73·7-s + 2.70·8-s + 5.34·9-s + 0.363·11-s − 3.88·12-s + 1.90·13-s − 3.02·14-s + 0.492·16-s + 1.56·17-s − 4.32·18-s − 4.74·19-s + 10.7·21-s − 0.294·22-s − 0.210·23-s + 7.82·24-s − 1.54·26-s + 6.76·27-s − 5.01·28-s + 3.72·29-s + 2.95·31-s − 5.81·32-s + 1.05·33-s − 1.27·34-s + ⋯
L(s)  = 1  − 0.572·2-s + 1.66·3-s − 0.671·4-s − 0.955·6-s + 1.41·7-s + 0.957·8-s + 1.78·9-s + 0.109·11-s − 1.12·12-s + 0.529·13-s − 0.808·14-s + 0.123·16-s + 0.380·17-s − 1.02·18-s − 1.08·19-s + 2.35·21-s − 0.0628·22-s − 0.0439·23-s + 1.59·24-s − 0.303·26-s + 1.30·27-s − 0.948·28-s + 0.691·29-s + 0.530·31-s − 1.02·32-s + 0.182·33-s − 0.217·34-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

\(L(1)\) \(\approx\) \(3.125042089\)
\(L(\frac12)\) \(\approx\) \(3.125042089\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.810T + 2T^{2} \)
3 \( 1 - 2.88T + 3T^{2} \)
7 \( 1 - 3.73T + 7T^{2} \)
11 \( 1 - 0.363T + 11T^{2} \)
13 \( 1 - 1.90T + 13T^{2} \)
17 \( 1 - 1.56T + 17T^{2} \)
19 \( 1 + 4.74T + 19T^{2} \)
23 \( 1 + 0.210T + 23T^{2} \)
29 \( 1 - 3.72T + 29T^{2} \)
31 \( 1 - 2.95T + 31T^{2} \)
37 \( 1 + 2.19T + 37T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 - 5.79T + 43T^{2} \)
47 \( 1 + 3.70T + 47T^{2} \)
53 \( 1 + 2.34T + 53T^{2} \)
59 \( 1 + 5.13T + 59T^{2} \)
61 \( 1 - 3.62T + 61T^{2} \)
67 \( 1 + 1.38T + 67T^{2} \)
71 \( 1 + 0.880T + 71T^{2} \)
73 \( 1 - 6.39T + 73T^{2} \)
79 \( 1 + 0.286T + 79T^{2} \)
83 \( 1 - 6.19T + 83T^{2} \)
89 \( 1 + 2.54T + 89T^{2} \)
97 \( 1 + 8.23T + 97T^{2} \)
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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

−8.214111017526533456931606494915, −7.83301349136588941746458631634, −7.10426297553845508689548453901, −5.97400055067253829904881661135, −4.86449529974524672008338777424, −4.32341012039576309621641571762, −3.70966239463953820986687659347, −2.62676880094790247928182435604, −1.80146569363211566788881294825, −1.03459520022394156215802882991, 1.03459520022394156215802882991, 1.80146569363211566788881294825, 2.62676880094790247928182435604, 3.70966239463953820986687659347, 4.32341012039576309621641571762, 4.86449529974524672008338777424, 5.97400055067253829904881661135, 7.10426297553845508689548453901, 7.83301349136588941746458631634, 8.214111017526533456931606494915

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