Properties

Label 2-5054-1.1-c1-0-133
Degree $2$
Conductor $5054$
Sign $-1$
Analytic cond. $40.3563$
Root an. cond. $6.35266$
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 − 2.14·3-s + 4-s + 2.45·5-s + 2.14·6-s + 7-s − 8-s + 1.60·9-s − 2.45·10-s + 4.66·11-s − 2.14·12-s + 1.96·13-s − 14-s − 5.26·15-s + 16-s − 1.67·17-s − 1.60·18-s + 2.45·20-s − 2.14·21-s − 4.66·22-s + 0.371·23-s + 2.14·24-s + 1.01·25-s − 1.96·26-s + 3.00·27-s + 28-s − 6.25·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.23·3-s + 0.5·4-s + 1.09·5-s + 0.875·6-s + 0.377·7-s − 0.353·8-s + 0.533·9-s − 0.775·10-s + 1.40·11-s − 0.619·12-s + 0.546·13-s − 0.267·14-s − 1.35·15-s + 0.250·16-s − 0.407·17-s − 0.377·18-s + 0.548·20-s − 0.468·21-s − 0.993·22-s + 0.0773·23-s + 0.437·24-s + 0.203·25-s − 0.386·26-s + 0.577·27-s + 0.188·28-s − 1.16·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5054\)    =    \(2 \cdot 7 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(40.3563\)
Root analytic conductor: \(6.35266\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5054} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5054,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
7 \( 1 - T \)
19 \( 1 \)
good3 \( 1 + 2.14T + 3T^{2} \)
5 \( 1 - 2.45T + 5T^{2} \)
11 \( 1 - 4.66T + 11T^{2} \)
13 \( 1 - 1.96T + 13T^{2} \)
17 \( 1 + 1.67T + 17T^{2} \)
23 \( 1 - 0.371T + 23T^{2} \)
29 \( 1 + 6.25T + 29T^{2} \)
31 \( 1 + 7.32T + 31T^{2} \)
37 \( 1 - 0.646T + 37T^{2} \)
41 \( 1 - 1.10T + 41T^{2} \)
43 \( 1 + 9.92T + 43T^{2} \)
47 \( 1 + 12.6T + 47T^{2} \)
53 \( 1 + 5.55T + 53T^{2} \)
59 \( 1 + 2.02T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 6.75T + 67T^{2} \)
71 \( 1 + 8.28T + 71T^{2} \)
73 \( 1 + 2.51T + 73T^{2} \)
79 \( 1 + 9.58T + 79T^{2} \)
83 \( 1 - 9.73T + 83T^{2} \)
89 \( 1 - 1.96T + 89T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−7.897180289071156284304031969322, −6.84933778786212944799807441046, −6.47488974074484787373644879021, −5.81622986732728877465323957806, −5.26238033404640770254939869533, −4.28436804831562626579064699806, −3.23658438021552824838533835799, −1.78961552068974456284501372531, −1.42697099610727802334058262190, 0, 1.42697099610727802334058262190, 1.78961552068974456284501372531, 3.23658438021552824838533835799, 4.28436804831562626579064699806, 5.26238033404640770254939869533, 5.81622986732728877465323957806, 6.47488974074484787373644879021, 6.84933778786212944799807441046, 7.897180289071156284304031969322

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