Properties

Label 2-2925-1.1-c1-0-47
Degree $2$
Conductor $2925$
Sign $-1$
Analytic cond. $23.3562$
Root an. cond. $4.83282$
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  − 0.634·2-s − 1.59·4-s − 2.74·7-s + 2.28·8-s − 4.00·11-s + 13-s + 1.74·14-s + 1.74·16-s + 2.35·17-s + 4.69·19-s + 2.54·22-s + 1.88·23-s − 0.634·26-s + 4.39·28-s + 3.19·31-s − 5.67·32-s − 1.49·34-s − 6.44·37-s − 2.97·38-s + 11.5·41-s + 0.691·43-s + 6.40·44-s − 1.19·46-s − 5.89·47-s + 0.552·49-s − 1.59·52-s + 9.14·53-s + ⋯
L(s)  = 1  − 0.448·2-s − 0.798·4-s − 1.03·7-s + 0.806·8-s − 1.20·11-s + 0.277·13-s + 0.465·14-s + 0.437·16-s + 0.572·17-s + 1.07·19-s + 0.541·22-s + 0.393·23-s − 0.124·26-s + 0.829·28-s + 0.573·31-s − 1.00·32-s − 0.256·34-s − 1.05·37-s − 0.482·38-s + 1.80·41-s + 0.105·43-s + 0.965·44-s − 0.176·46-s − 0.859·47-s + 0.0789·49-s − 0.221·52-s + 1.25·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(23.3562\)
Root analytic conductor: \(4.83282\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2925,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 - T \)
good2 \( 1 + 0.634T + 2T^{2} \)
7 \( 1 + 2.74T + 7T^{2} \)
11 \( 1 + 4.00T + 11T^{2} \)
17 \( 1 - 2.35T + 17T^{2} \)
19 \( 1 - 4.69T + 19T^{2} \)
23 \( 1 - 1.88T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 3.19T + 31T^{2} \)
37 \( 1 + 6.44T + 37T^{2} \)
41 \( 1 - 11.5T + 41T^{2} \)
43 \( 1 - 0.691T + 43T^{2} \)
47 \( 1 + 5.89T + 47T^{2} \)
53 \( 1 - 9.14T + 53T^{2} \)
59 \( 1 - 1.17T + 59T^{2} \)
61 \( 1 + 4.44T + 61T^{2} \)
67 \( 1 + 4.80T + 67T^{2} \)
71 \( 1 - 1.82T + 71T^{2} \)
73 \( 1 + 15.3T + 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 + 8.42T + 83T^{2} \)
89 \( 1 + 5.75T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−8.484401251821378861154016836758, −7.66034127625577950017181173873, −7.13811276393380168651029618346, −5.94844875184154030825830465661, −5.36269270719890828333500426071, −4.48241282904276873373670103430, −3.46008775977508702403999237501, −2.75670996760833194461812164664, −1.17481167966041735397887190724, 0, 1.17481167966041735397887190724, 2.75670996760833194461812164664, 3.46008775977508702403999237501, 4.48241282904276873373670103430, 5.36269270719890828333500426071, 5.94844875184154030825830465661, 7.13811276393380168651029618346, 7.66034127625577950017181173873, 8.484401251821378861154016836758

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