Properties

Label 2-2925-1.1-c1-0-36
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  + 0.329·2-s − 1.89·4-s + 3.70·7-s − 1.28·8-s + 3.31·11-s + 13-s + 1.21·14-s + 3.36·16-s + 4.36·17-s + 5.21·19-s + 1.09·22-s − 4.92·23-s + 0.329·26-s − 7.00·28-s − 7.78·29-s + 0.0981·31-s + 3.67·32-s + 1.43·34-s − 2.92·37-s + 1.71·38-s + 0.749·41-s − 3.78·43-s − 6.26·44-s − 1.62·46-s − 5.67·47-s + 6.70·49-s − 1.89·52-s + ⋯
L(s)  = 1  + 0.232·2-s − 0.945·4-s + 1.39·7-s − 0.453·8-s + 0.998·11-s + 0.277·13-s + 0.325·14-s + 0.840·16-s + 1.05·17-s + 1.19·19-s + 0.232·22-s − 1.02·23-s + 0.0645·26-s − 1.32·28-s − 1.44·29-s + 0.0176·31-s + 0.648·32-s + 0.246·34-s − 0.480·37-s + 0.278·38-s + 0.117·41-s − 0.576·43-s − 0.944·44-s − 0.239·46-s − 0.827·47-s + 0.957·49-s − 0.262·52-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

\(L(1)\) \(\approx\) \(2.167968876\)
\(L(\frac12)\) \(\approx\) \(2.167968876\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.329T + 2T^{2} \)
7 \( 1 - 3.70T + 7T^{2} \)
11 \( 1 - 3.31T + 11T^{2} \)
17 \( 1 - 4.36T + 17T^{2} \)
19 \( 1 - 5.21T + 19T^{2} \)
23 \( 1 + 4.92T + 23T^{2} \)
29 \( 1 + 7.78T + 29T^{2} \)
31 \( 1 - 0.0981T + 31T^{2} \)
37 \( 1 + 2.92T + 37T^{2} \)
41 \( 1 - 0.749T + 41T^{2} \)
43 \( 1 + 3.78T + 43T^{2} \)
47 \( 1 + 5.67T + 47T^{2} \)
53 \( 1 - 2.19T + 53T^{2} \)
59 \( 1 + 0.108T + 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 + 12.0T + 71T^{2} \)
73 \( 1 - 9.56T + 73T^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 - 16.7T + 83T^{2} \)
89 \( 1 - 3.59T + 89T^{2} \)
97 \( 1 + 4.10T + 97T^{2} \)
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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

−8.705799103307122607493416799051, −8.019755162821666098595536100221, −7.51004798055550713606956277931, −6.32860457339978588714156903478, −5.38314110617990692057891923107, −5.03363597708576973570099901411, −3.92909756296133098139700492703, −3.50491927746252644374413556630, −1.87572447660702369894411644694, −0.955795630899186280450774250351, 0.955795630899186280450774250351, 1.87572447660702369894411644694, 3.50491927746252644374413556630, 3.92909756296133098139700492703, 5.03363597708576973570099901411, 5.38314110617990692057891923107, 6.32860457339978588714156903478, 7.51004798055550713606956277931, 8.019755162821666098595536100221, 8.705799103307122607493416799051

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