Properties

Label 2-4600-1.1-c1-0-71
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  − 1.83·3-s + 3.97·7-s + 0.384·9-s + 2.10·11-s − 5.35·13-s − 1.29·17-s + 2.10·19-s − 7.30·21-s + 23-s + 4.81·27-s + 6.03·29-s − 8.32·31-s − 3.86·33-s − 5.10·37-s + 9.85·39-s − 8.33·41-s − 7.78·43-s − 11.3·47-s + 8.78·49-s + 2.38·51-s + 0.573·53-s − 3.86·57-s − 9.17·59-s + 13.5·61-s + 1.52·63-s + 15.8·67-s − 1.83·69-s + ⋯
L(s)  = 1  − 1.06·3-s + 1.50·7-s + 0.128·9-s + 0.634·11-s − 1.48·13-s − 0.314·17-s + 0.482·19-s − 1.59·21-s + 0.208·23-s + 0.926·27-s + 1.11·29-s − 1.49·31-s − 0.673·33-s − 0.838·37-s + 1.57·39-s − 1.30·41-s − 1.18·43-s − 1.66·47-s + 1.25·49-s + 0.333·51-s + 0.0787·53-s − 0.512·57-s − 1.19·59-s + 1.73·61-s + 0.192·63-s + 1.93·67-s − 0.221·69-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 \( 1 + 1.83T + 3T^{2} \)
7 \( 1 - 3.97T + 7T^{2} \)
11 \( 1 - 2.10T + 11T^{2} \)
13 \( 1 + 5.35T + 13T^{2} \)
17 \( 1 + 1.29T + 17T^{2} \)
19 \( 1 - 2.10T + 19T^{2} \)
29 \( 1 - 6.03T + 29T^{2} \)
31 \( 1 + 8.32T + 31T^{2} \)
37 \( 1 + 5.10T + 37T^{2} \)
41 \( 1 + 8.33T + 41T^{2} \)
43 \( 1 + 7.78T + 43T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 - 0.573T + 53T^{2} \)
59 \( 1 + 9.17T + 59T^{2} \)
61 \( 1 - 13.5T + 61T^{2} \)
67 \( 1 - 15.8T + 67T^{2} \)
71 \( 1 - 14.9T + 71T^{2} \)
73 \( 1 + 8.36T + 73T^{2} \)
79 \( 1 + 9.41T + 79T^{2} \)
83 \( 1 + 1.51T + 83T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 - 0.337T + 97T^{2} \)
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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

−8.028093372630654394854279885579, −6.97690040483371983793553661535, −6.69493486074987815065934749684, −5.44002204580258328839138768248, −5.09378947524076406896767298661, −4.58365262694259645078219343723, −3.42197564556176660376842552779, −2.19439798856536551129031234110, −1.32134206307823997720091627963, 0, 1.32134206307823997720091627963, 2.19439798856536551129031234110, 3.42197564556176660376842552779, 4.58365262694259645078219343723, 5.09378947524076406896767298661, 5.44002204580258328839138768248, 6.69493486074987815065934749684, 6.97690040483371983793553661535, 8.028093372630654394854279885579

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