Properties

Label 2-7400-1.1-c1-0-69
Degree $2$
Conductor $7400$
Sign $1$
Analytic cond. $59.0892$
Root an. cond. $7.68695$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·7-s + 9-s + 6·13-s + 2·17-s + 6·19-s − 4·21-s + 4·23-s − 4·27-s − 2·29-s − 6·31-s − 37-s + 12·39-s − 6·41-s − 8·43-s + 6·47-s − 3·49-s + 4·51-s + 10·53-s + 12·57-s + 14·59-s + 6·61-s − 2·63-s + 14·67-s + 8·69-s + 8·71-s + 6·73-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.755·7-s + 1/3·9-s + 1.66·13-s + 0.485·17-s + 1.37·19-s − 0.872·21-s + 0.834·23-s − 0.769·27-s − 0.371·29-s − 1.07·31-s − 0.164·37-s + 1.92·39-s − 0.937·41-s − 1.21·43-s + 0.875·47-s − 3/7·49-s + 0.560·51-s + 1.37·53-s + 1.58·57-s + 1.82·59-s + 0.768·61-s − 0.251·63-s + 1.71·67-s + 0.963·69-s + 0.949·71-s + 0.702·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7400\)    =    \(2^{3} \cdot 5^{2} \cdot 37\)
Sign: $1$
Analytic conductor: \(59.0892\)
Root analytic conductor: \(7.68695\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.206112950\)
\(L(\frac12)\) \(\approx\) \(3.206112950\)
\(L(\frac{3}{2})\) 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 \( 1 \)
37 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.074891635766676026053842589464, −7.19727611471581005982526437649, −6.72717722909364069619007956497, −5.69028796919509436905256947949, −5.27532115821775196687730474628, −3.80425817796001687024547559149, −3.56136131373462145924103665881, −2.91884113801564677170571066690, −1.89090821102870243826723745948, −0.874661025439769712868759713560, 0.874661025439769712868759713560, 1.89090821102870243826723745948, 2.91884113801564677170571066690, 3.56136131373462145924103665881, 3.80425817796001687024547559149, 5.27532115821775196687730474628, 5.69028796919509436905256947949, 6.72717722909364069619007956497, 7.19727611471581005982526437649, 8.074891635766676026053842589464

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