Properties

Label 2-70e2-1.1-c1-0-61
Degree $2$
Conductor $4900$
Sign $-1$
Analytic cond. $39.1266$
Root an. cond. $6.25513$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 9-s − 11-s + 2·13-s − 4·17-s − 5·23-s − 4·27-s − 3·29-s − 10·31-s − 2·33-s − 5·37-s + 4·39-s − 10·41-s + 5·43-s + 4·47-s − 8·51-s − 10·53-s − 10·59-s + 10·61-s − 5·67-s − 10·69-s + 3·71-s + 10·73-s + 13·79-s − 11·81-s + 10·83-s − 6·87-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.970·17-s − 1.04·23-s − 0.769·27-s − 0.557·29-s − 1.79·31-s − 0.348·33-s − 0.821·37-s + 0.640·39-s − 1.56·41-s + 0.762·43-s + 0.583·47-s − 1.12·51-s − 1.37·53-s − 1.30·59-s + 1.28·61-s − 0.610·67-s − 1.20·69-s + 0.356·71-s + 1.17·73-s + 1.46·79-s − 1.22·81-s + 1.09·83-s − 0.643·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(39.1266\)
Root analytic conductor: \(6.25513\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4900} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4900,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−8.003517164822898453706522737855, −7.38347489855123033375281405421, −6.54035550094457916590735795067, −5.73634146260854394987137444257, −4.89417069155550754897518684774, −3.81842598982803925176332182024, −3.44051038886752138816370390671, −2.32278285635521864702487030361, −1.75869986356759446351281344214, 0, 1.75869986356759446351281344214, 2.32278285635521864702487030361, 3.44051038886752138816370390671, 3.81842598982803925176332182024, 4.89417069155550754897518684774, 5.73634146260854394987137444257, 6.54035550094457916590735795067, 7.38347489855123033375281405421, 8.003517164822898453706522737855

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