Properties

Label 2-608-1.1-c3-0-5
Degree $2$
Conductor $608$
Sign $1$
Analytic cond. $35.8731$
Root an. cond. $5.98942$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.43·3-s − 9.42·5-s − 9.58·7-s − 21.0·9-s − 37.8·11-s − 30.7·13-s + 22.9·15-s + 35.3·17-s − 19·19-s + 23.3·21-s − 88.7·23-s − 36.1·25-s + 117.·27-s − 226.·29-s + 320.·31-s + 92.2·33-s + 90.3·35-s + 346.·37-s + 75.0·39-s − 150.·41-s − 284.·43-s + 198.·45-s + 240.·47-s − 251.·49-s − 86.1·51-s − 539.·53-s + 356.·55-s + ⋯
L(s)  = 1  − 0.469·3-s − 0.843·5-s − 0.517·7-s − 0.779·9-s − 1.03·11-s − 0.656·13-s + 0.395·15-s + 0.504·17-s − 0.229·19-s + 0.242·21-s − 0.804·23-s − 0.289·25-s + 0.835·27-s − 1.44·29-s + 1.85·31-s + 0.486·33-s + 0.436·35-s + 1.54·37-s + 0.308·39-s − 0.574·41-s − 1.00·43-s + 0.657·45-s + 0.746·47-s − 0.731·49-s − 0.236·51-s − 1.39·53-s + 0.875·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $1$
Analytic conductor: \(35.8731\)
Root analytic conductor: \(5.98942\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4949991439\)
\(L(\frac12)\) \(\approx\) \(0.4949991439\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + 19T \)
good3 \( 1 + 2.43T + 27T^{2} \)
5 \( 1 + 9.42T + 125T^{2} \)
7 \( 1 + 9.58T + 343T^{2} \)
11 \( 1 + 37.8T + 1.33e3T^{2} \)
13 \( 1 + 30.7T + 2.19e3T^{2} \)
17 \( 1 - 35.3T + 4.91e3T^{2} \)
23 \( 1 + 88.7T + 1.21e4T^{2} \)
29 \( 1 + 226.T + 2.43e4T^{2} \)
31 \( 1 - 320.T + 2.97e4T^{2} \)
37 \( 1 - 346.T + 5.06e4T^{2} \)
41 \( 1 + 150.T + 6.89e4T^{2} \)
43 \( 1 + 284.T + 7.95e4T^{2} \)
47 \( 1 - 240.T + 1.03e5T^{2} \)
53 \( 1 + 539.T + 1.48e5T^{2} \)
59 \( 1 - 889.T + 2.05e5T^{2} \)
61 \( 1 + 377.T + 2.26e5T^{2} \)
67 \( 1 + 78.4T + 3.00e5T^{2} \)
71 \( 1 - 924.T + 3.57e5T^{2} \)
73 \( 1 - 576.T + 3.89e5T^{2} \)
79 \( 1 - 601.T + 4.93e5T^{2} \)
83 \( 1 + 1.02e3T + 5.71e5T^{2} \)
89 \( 1 - 828.T + 7.04e5T^{2} \)
97 \( 1 + 101.T + 9.12e5T^{2} \)
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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

−10.25375879453299936445419366748, −9.542441971536762287002780115224, −8.176769454259524307731654588358, −7.80856525428186086189740901439, −6.57570798824549143722115528770, −5.66948202737277631591980335738, −4.73482920154314224160624157785, −3.51913146047258069674004208198, −2.46541778660233114366889952555, −0.40106291228761212795795795271, 0.40106291228761212795795795271, 2.46541778660233114366889952555, 3.51913146047258069674004208198, 4.73482920154314224160624157785, 5.66948202737277631591980335738, 6.57570798824549143722115528770, 7.80856525428186086189740901439, 8.176769454259524307731654588358, 9.542441971536762287002780115224, 10.25375879453299936445419366748

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