Properties

Label 2-2925-1.1-c1-0-26
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  − 2.16·2-s + 2.68·4-s + 3.26·7-s − 1.47·8-s + 2.85·11-s − 13-s − 7.07·14-s − 2.17·16-s − 6.38·17-s + 3.68·19-s − 6.17·22-s − 5.01·23-s + 2.16·26-s + 8.76·28-s + 2.05·29-s + 5.58·31-s + 7.65·32-s + 13.8·34-s + 10.2·37-s − 7.96·38-s + 5.01·41-s + 0.319·43-s + 7.65·44-s + 10.8·46-s + 2.16·47-s + 3.68·49-s − 2.68·52-s + ⋯
L(s)  = 1  − 1.52·2-s + 1.34·4-s + 1.23·7-s − 0.520·8-s + 0.860·11-s − 0.277·13-s − 1.88·14-s − 0.543·16-s − 1.54·17-s + 0.844·19-s − 1.31·22-s − 1.04·23-s + 0.424·26-s + 1.65·28-s + 0.381·29-s + 1.00·31-s + 1.35·32-s + 2.36·34-s + 1.67·37-s − 1.29·38-s + 0.783·41-s + 0.0486·43-s + 1.15·44-s + 1.60·46-s + 0.315·47-s + 0.525·49-s − 0.371·52-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

\(L(1)\) \(\approx\) \(0.9934311868\)
\(L(\frac12)\) \(\approx\) \(0.9934311868\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.16T + 2T^{2} \)
7 \( 1 - 3.26T + 7T^{2} \)
11 \( 1 - 2.85T + 11T^{2} \)
17 \( 1 + 6.38T + 17T^{2} \)
19 \( 1 - 3.68T + 19T^{2} \)
23 \( 1 + 5.01T + 23T^{2} \)
29 \( 1 - 2.05T + 29T^{2} \)
31 \( 1 - 5.58T + 31T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 - 5.01T + 41T^{2} \)
43 \( 1 - 0.319T + 43T^{2} \)
47 \( 1 - 2.16T + 47T^{2} \)
53 \( 1 + 2.96T + 53T^{2} \)
59 \( 1 + 6.49T + 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 + 7.58T + 67T^{2} \)
71 \( 1 - 16.3T + 71T^{2} \)
73 \( 1 + 5.04T + 73T^{2} \)
79 \( 1 - 0.957T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + 7.05T + 89T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−8.702268455727458272833246623566, −8.152285645579024641709101190231, −7.54924376416428839710061838825, −6.76860406574090731593945398140, −5.99506500068837283668416467561, −4.71498386244825420370765438662, −4.20146438877014732153910762963, −2.57904601776757049236965364466, −1.76153612000146880303260058121, −0.805704281633056796462299086766, 0.805704281633056796462299086766, 1.76153612000146880303260058121, 2.57904601776757049236965364466, 4.20146438877014732153910762963, 4.71498386244825420370765438662, 5.99506500068837283668416467561, 6.76860406574090731593945398140, 7.54924376416428839710061838825, 8.152285645579024641709101190231, 8.702268455727458272833246623566

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