Properties

Label 2-4400-1.1-c1-0-85
Degree $2$
Conductor $4400$
Sign $-1$
Analytic cond. $35.1341$
Root an. cond. $5.92740$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.61·3-s − 2.85·7-s + 3.85·9-s − 11-s − 6.23·13-s + 0.618·17-s + 6.70·19-s − 7.47·21-s − 4.09·23-s + 2.23·27-s − 1.38·29-s + 3·31-s − 2.61·33-s − 10.2·37-s − 16.3·39-s − 3·41-s − 6·43-s + 11.9·47-s + 1.14·49-s + 1.61·51-s − 9.32·53-s + 17.5·57-s − 0.527·59-s + 0.0901·61-s − 11.0·63-s + 8·67-s − 10.7·69-s + ⋯
L(s)  = 1  + 1.51·3-s − 1.07·7-s + 1.28·9-s − 0.301·11-s − 1.72·13-s + 0.149·17-s + 1.53·19-s − 1.63·21-s − 0.852·23-s + 0.430·27-s − 0.256·29-s + 0.538·31-s − 0.455·33-s − 1.68·37-s − 2.61·39-s − 0.468·41-s − 0.914·43-s + 1.74·47-s + 0.163·49-s + 0.226·51-s − 1.28·53-s + 2.32·57-s − 0.0687·59-s + 0.0115·61-s − 1.38·63-s + 0.977·67-s − 1.28·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4400\)    =    \(2^{4} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(35.1341\)
Root analytic conductor: \(5.92740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 + T \)
good3 \( 1 - 2.61T + 3T^{2} \)
7 \( 1 + 2.85T + 7T^{2} \)
13 \( 1 + 6.23T + 13T^{2} \)
17 \( 1 - 0.618T + 17T^{2} \)
19 \( 1 - 6.70T + 19T^{2} \)
23 \( 1 + 4.09T + 23T^{2} \)
29 \( 1 + 1.38T + 29T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 - 11.9T + 47T^{2} \)
53 \( 1 + 9.32T + 53T^{2} \)
59 \( 1 + 0.527T + 59T^{2} \)
61 \( 1 - 0.0901T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 8.18T + 71T^{2} \)
73 \( 1 + 10.3T + 73T^{2} \)
79 \( 1 + 5.85T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + 6.90T + 89T^{2} \)
97 \( 1 + 1.61T + 97T^{2} \)
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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

−7.968275165552492968490092800514, −7.35428668053940891074640686375, −6.90522295290704701884676870667, −5.74835161620224278970938375701, −4.95004495714196614097970491129, −3.93777399207267259103718415756, −3.11576859801955217031266819624, −2.73017203621577420589298247557, −1.72256025701593594240909040968, 0, 1.72256025701593594240909040968, 2.73017203621577420589298247557, 3.11576859801955217031266819624, 3.93777399207267259103718415756, 4.95004495714196614097970491129, 5.74835161620224278970938375701, 6.90522295290704701884676870667, 7.35428668053940891074640686375, 7.968275165552492968490092800514

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