Properties

Label 2-2925-1.1-c1-0-30
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.56·2-s + 0.438·4-s + 4.56·7-s + 2.43·8-s − 2.56·11-s + 13-s − 7.12·14-s − 4.68·16-s − 2.56·17-s + 3.12·19-s + 4·22-s + 6.56·23-s − 1.56·26-s + 1.99·28-s − 1.12·29-s + 6·31-s + 2.43·32-s + 4·34-s − 1.68·37-s − 4.87·38-s + 0.561·41-s + 5.12·43-s − 1.12·44-s − 10.2·46-s + 2.87·47-s + 13.8·49-s + 0.438·52-s + ⋯
L(s)  = 1  − 1.10·2-s + 0.219·4-s + 1.72·7-s + 0.862·8-s − 0.772·11-s + 0.277·13-s − 1.90·14-s − 1.17·16-s − 0.621·17-s + 0.716·19-s + 0.852·22-s + 1.36·23-s − 0.306·26-s + 0.377·28-s − 0.208·29-s + 1.07·31-s + 0.431·32-s + 0.685·34-s − 0.276·37-s − 0.791·38-s + 0.0876·41-s + 0.781·43-s − 0.169·44-s − 1.51·46-s + 0.419·47-s + 1.97·49-s + 0.0608·52-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: \(0\)
Selberg data: \((2,\ 2925,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.187960663\)
\(L(\frac12)\) \(\approx\) \(1.187960663\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.56T + 2T^{2} \)
7 \( 1 - 4.56T + 7T^{2} \)
11 \( 1 + 2.56T + 11T^{2} \)
17 \( 1 + 2.56T + 17T^{2} \)
19 \( 1 - 3.12T + 19T^{2} \)
23 \( 1 - 6.56T + 23T^{2} \)
29 \( 1 + 1.12T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + 1.68T + 37T^{2} \)
41 \( 1 - 0.561T + 41T^{2} \)
43 \( 1 - 5.12T + 43T^{2} \)
47 \( 1 - 2.87T + 47T^{2} \)
53 \( 1 - 7.68T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 5.68T + 61T^{2} \)
67 \( 1 + 13.3T + 67T^{2} \)
71 \( 1 + 14.5T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 7.68T + 79T^{2} \)
83 \( 1 - 16.4T + 83T^{2} \)
89 \( 1 - 1.68T + 89T^{2} \)
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

−8.921704959136110411258854096121, −7.921175777566775559343205342529, −7.70413487995788049060092879866, −6.84069148812370687305857169408, −5.54929761566087106478187426473, −4.86259423558116738394237999120, −4.26371281707278115126355141378, −2.77107166179636076775149333695, −1.72491052929441478977119378052, −0.849040071871901970998004812377, 0.849040071871901970998004812377, 1.72491052929441478977119378052, 2.77107166179636076775149333695, 4.26371281707278115126355141378, 4.86259423558116738394237999120, 5.54929761566087106478187426473, 6.84069148812370687305857169408, 7.70413487995788049060092879866, 7.921175777566775559343205342529, 8.921704959136110411258854096121

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