Properties

Label 2-4600-1.1-c1-0-74
Degree $2$
Conductor $4600$
Sign $-1$
Analytic cond. $36.7311$
Root an. cond. $6.06062$
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  − 1.40·3-s + 1.57·7-s − 1.03·9-s + 4.35·11-s + 0.964·13-s − 0.300·17-s − 8.62·19-s − 2.20·21-s − 23-s + 5.65·27-s + 4.76·29-s − 5.59·31-s − 6.11·33-s − 4.38·37-s − 1.35·39-s − 6.62·41-s − 1.72·43-s − 0.687·47-s − 4.53·49-s + 0.421·51-s + 8.05·53-s + 12.1·57-s + 5.74·59-s − 13.6·61-s − 1.61·63-s + 6.49·67-s + 1.40·69-s + ⋯
L(s)  = 1  − 0.810·3-s + 0.593·7-s − 0.343·9-s + 1.31·11-s + 0.267·13-s − 0.0728·17-s − 1.97·19-s − 0.481·21-s − 0.208·23-s + 1.08·27-s + 0.885·29-s − 1.00·31-s − 1.06·33-s − 0.720·37-s − 0.216·39-s − 1.03·41-s − 0.263·43-s − 0.100·47-s − 0.647·49-s + 0.0589·51-s + 1.10·53-s + 1.60·57-s + 0.748·59-s − 1.74·61-s − 0.204·63-s + 0.792·67-s + 0.168·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4600\)    =    \(2^{3} \cdot 5^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(36.7311\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
23 \( 1 + T \)
good3 \( 1 + 1.40T + 3T^{2} \)
7 \( 1 - 1.57T + 7T^{2} \)
11 \( 1 - 4.35T + 11T^{2} \)
13 \( 1 - 0.964T + 13T^{2} \)
17 \( 1 + 0.300T + 17T^{2} \)
19 \( 1 + 8.62T + 19T^{2} \)
29 \( 1 - 4.76T + 29T^{2} \)
31 \( 1 + 5.59T + 31T^{2} \)
37 \( 1 + 4.38T + 37T^{2} \)
41 \( 1 + 6.62T + 41T^{2} \)
43 \( 1 + 1.72T + 43T^{2} \)
47 \( 1 + 0.687T + 47T^{2} \)
53 \( 1 - 8.05T + 53T^{2} \)
59 \( 1 - 5.74T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 - 6.49T + 67T^{2} \)
71 \( 1 - 9.89T + 71T^{2} \)
73 \( 1 - 6.35T + 73T^{2} \)
79 \( 1 + 6.95T + 79T^{2} \)
83 \( 1 + 0.185T + 83T^{2} \)
89 \( 1 + 1.64T + 89T^{2} \)
97 \( 1 + 9.21T + 97T^{2} \)
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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

−8.195452153248291171945600306839, −6.88402269584356254649447322659, −6.55606761426297950264096842143, −5.81895054387446334082251161787, −5.01851818462006056966813788145, −4.28815628373306400820254355716, −3.52878356067452232782938261618, −2.23758334698004661215997212622, −1.32379314990075453694803910124, 0, 1.32379314990075453694803910124, 2.23758334698004661215997212622, 3.52878356067452232782938261618, 4.28815628373306400820254355716, 5.01851818462006056966813788145, 5.81895054387446334082251161787, 6.55606761426297950264096842143, 6.88402269584356254649447322659, 8.195452153248291171945600306839

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