Properties

Label 2-10e2-1.1-c3-0-0
Degree $2$
Conductor $100$
Sign $1$
Analytic cond. $5.90019$
Root an. cond. $2.42903$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.71·3-s − 8.71·7-s + 49.0·9-s + 20·11-s + 52.3·13-s + 69.7·17-s + 84·19-s + 76.0·21-s − 61.0·23-s − 191.·27-s − 6·29-s − 224·31-s − 174.·33-s + 122.·37-s − 456.·39-s + 266·41-s + 305.·43-s + 374.·47-s − 267·49-s − 608.·51-s + 366.·53-s − 732.·57-s + 28·59-s + 182·61-s − 427.·63-s + 427.·67-s + 532.·69-s + ⋯
L(s)  = 1  − 1.67·3-s − 0.470·7-s + 1.81·9-s + 0.548·11-s + 1.11·13-s + 0.995·17-s + 1.01·19-s + 0.789·21-s − 0.553·23-s − 1.36·27-s − 0.0384·29-s − 1.29·31-s − 0.919·33-s + 0.542·37-s − 1.87·39-s + 1.01·41-s + 1.08·43-s + 1.16·47-s − 0.778·49-s − 1.66·51-s + 0.948·53-s − 1.70·57-s + 0.0617·59-s + 0.382·61-s − 0.854·63-s + 0.778·67-s + 0.928·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(5.90019\)
Root analytic conductor: \(2.42903\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8977179166\)
\(L(\frac12)\) \(\approx\) \(0.8977179166\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + 8.71T + 27T^{2} \)
7 \( 1 + 8.71T + 343T^{2} \)
11 \( 1 - 20T + 1.33e3T^{2} \)
13 \( 1 - 52.3T + 2.19e3T^{2} \)
17 \( 1 - 69.7T + 4.91e3T^{2} \)
19 \( 1 - 84T + 6.85e3T^{2} \)
23 \( 1 + 61.0T + 1.21e4T^{2} \)
29 \( 1 + 6T + 2.43e4T^{2} \)
31 \( 1 + 224T + 2.97e4T^{2} \)
37 \( 1 - 122.T + 5.06e4T^{2} \)
41 \( 1 - 266T + 6.89e4T^{2} \)
43 \( 1 - 305.T + 7.95e4T^{2} \)
47 \( 1 - 374.T + 1.03e5T^{2} \)
53 \( 1 - 366.T + 1.48e5T^{2} \)
59 \( 1 - 28T + 2.05e5T^{2} \)
61 \( 1 - 182T + 2.26e5T^{2} \)
67 \( 1 - 427.T + 3.00e5T^{2} \)
71 \( 1 - 408T + 3.57e5T^{2} \)
73 \( 1 + 1.08e3T + 3.89e5T^{2} \)
79 \( 1 + 48T + 4.93e5T^{2} \)
83 \( 1 - 200.T + 5.71e5T^{2} \)
89 \( 1 - 1.52e3T + 7.04e5T^{2} \)
97 \( 1 - 557.T + 9.12e5T^{2} \)
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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

−13.08526551722921863255049344935, −12.14539606027348282986297821432, −11.35397713553133950884991328701, −10.41984219251924784885355088362, −9.298269538310991669903440298132, −7.47526109559795748377691533915, −6.22834592740723297880775433461, −5.49095945825063381450787787157, −3.86005185487578236921236990744, −0.969702454606225205489518811946, 0.969702454606225205489518811946, 3.86005185487578236921236990744, 5.49095945825063381450787787157, 6.22834592740723297880775433461, 7.47526109559795748377691533915, 9.298269538310991669903440298132, 10.41984219251924784885355088362, 11.35397713553133950884991328701, 12.14539606027348282986297821432, 13.08526551722921863255049344935

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