Properties

Label 2-4368-1.1-c1-0-30
Degree $2$
Conductor $4368$
Sign $1$
Analytic cond. $34.8786$
Root an. cond. $5.90581$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2.70·5-s − 7-s + 9-s + 0.701·11-s + 13-s + 2.70·15-s − 2.70·17-s + 0.701·19-s − 21-s − 4.70·23-s + 2.29·25-s + 27-s + 2.70·29-s + 0.701·33-s − 2.70·35-s + 10.7·37-s + 39-s + 3.40·41-s + 10.1·43-s + 2.70·45-s + 8·47-s + 49-s − 2.70·51-s − 2·53-s + 1.89·55-s + 0.701·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.20·5-s − 0.377·7-s + 0.333·9-s + 0.211·11-s + 0.277·13-s + 0.697·15-s − 0.655·17-s + 0.160·19-s − 0.218·21-s − 0.980·23-s + 0.459·25-s + 0.192·27-s + 0.501·29-s + 0.122·33-s − 0.456·35-s + 1.75·37-s + 0.160·39-s + 0.531·41-s + 1.54·43-s + 0.402·45-s + 1.16·47-s + 0.142·49-s − 0.378·51-s − 0.274·53-s + 0.255·55-s + 0.0929·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4368\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(34.8786\)
Root analytic conductor: \(5.90581\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4368,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.072910929\)
\(L(\frac12)\) \(\approx\) \(3.072910929\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 - T \)
good5 \( 1 - 2.70T + 5T^{2} \)
11 \( 1 - 0.701T + 11T^{2} \)
17 \( 1 + 2.70T + 17T^{2} \)
19 \( 1 - 0.701T + 19T^{2} \)
23 \( 1 + 4.70T + 23T^{2} \)
29 \( 1 - 2.70T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 - 3.40T + 41T^{2} \)
43 \( 1 - 10.1T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 14.8T + 59T^{2} \)
61 \( 1 - 1.29T + 61T^{2} \)
67 \( 1 + 5.40T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 1.29T + 73T^{2} \)
79 \( 1 + 9.40T + 79T^{2} \)
83 \( 1 - 13.4T + 83T^{2} \)
89 \( 1 + 8.80T + 89T^{2} \)
97 \( 1 + 8.80T + 97T^{2} \)
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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

−8.475082915831101042953912842286, −7.68055848797064308405061146148, −6.85186836284450535489315301650, −6.08981399826918994713517293673, −5.67037557941313795329097237921, −4.49112396178094989313941162990, −3.82542256273838379028773987727, −2.64902807267948658255126067862, −2.17115554342392775260645601725, −0.990966220395811177725472490916, 0.990966220395811177725472490916, 2.17115554342392775260645601725, 2.64902807267948658255126067862, 3.82542256273838379028773987727, 4.49112396178094989313941162990, 5.67037557941313795329097237921, 6.08981399826918994713517293673, 6.85186836284450535489315301650, 7.68055848797064308405061146148, 8.475082915831101042953912842286

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