Properties

Label 2-608-1.1-c3-0-4
Degree $2$
Conductor $608$
Sign $1$
Analytic cond. $35.8731$
Root an. cond. $5.98942$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 8·5-s − 17·7-s − 26·9-s − 70·11-s − 61·13-s + 8·15-s + 83·17-s + 19·19-s + 17·21-s + 115·23-s − 61·25-s + 53·27-s + 279·29-s − 72·31-s + 70·33-s + 136·35-s − 34·37-s + 61·39-s + 108·41-s − 192·43-s + 208·45-s − 392·47-s − 54·49-s − 83·51-s + 131·53-s + 560·55-s + ⋯
L(s)  = 1  − 0.192·3-s − 0.715·5-s − 0.917·7-s − 0.962·9-s − 1.91·11-s − 1.30·13-s + 0.137·15-s + 1.18·17-s + 0.229·19-s + 0.176·21-s + 1.04·23-s − 0.487·25-s + 0.377·27-s + 1.78·29-s − 0.417·31-s + 0.369·33-s + 0.656·35-s − 0.151·37-s + 0.250·39-s + 0.411·41-s − 0.680·43-s + 0.689·45-s − 1.21·47-s − 0.157·49-s − 0.227·51-s + 0.339·53-s + 1.37·55-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5030716177\)
\(L(\frac12)\) \(\approx\) \(0.5030716177\)
\(L(\frac{5}{2})\) 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 \( 1 - p T \)
good3 \( 1 + T + p^{3} T^{2} \)
5 \( 1 + 8 T + p^{3} T^{2} \)
7 \( 1 + 17 T + p^{3} T^{2} \)
11 \( 1 + 70 T + p^{3} T^{2} \)
13 \( 1 + 61 T + p^{3} T^{2} \)
17 \( 1 - 83 T + p^{3} T^{2} \)
23 \( 1 - 5 p T + p^{3} T^{2} \)
29 \( 1 - 279 T + p^{3} T^{2} \)
31 \( 1 + 72 T + p^{3} T^{2} \)
37 \( 1 + 34 T + p^{3} T^{2} \)
41 \( 1 - 108 T + p^{3} T^{2} \)
43 \( 1 + 192 T + p^{3} T^{2} \)
47 \( 1 + 392 T + p^{3} T^{2} \)
53 \( 1 - 131 T + p^{3} T^{2} \)
59 \( 1 + 609 T + p^{3} T^{2} \)
61 \( 1 - 338 T + p^{3} T^{2} \)
67 \( 1 + 461 T + p^{3} T^{2} \)
71 \( 1 - 750 T + p^{3} T^{2} \)
73 \( 1 - 1177 T + p^{3} T^{2} \)
79 \( 1 + 22 T + p^{3} T^{2} \)
83 \( 1 + 810 T + p^{3} T^{2} \)
89 \( 1 + 476 T + p^{3} T^{2} \)
97 \( 1 - 1426 T + p^{3} T^{2} \)
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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

−10.22067562738484038150612112691, −9.555401770216505775026403185362, −8.240591487742029209996247990909, −7.72763477029040706421268850312, −6.72147970292783061993745307030, −5.49381665751844434923048690668, −4.89172663995033909533676922454, −3.23941468403453953930574411624, −2.69246169767261241422668699588, −0.39541448696797579928230028393, 0.39541448696797579928230028393, 2.69246169767261241422668699588, 3.23941468403453953930574411624, 4.89172663995033909533676922454, 5.49381665751844434923048690668, 6.72147970292783061993745307030, 7.72763477029040706421268850312, 8.240591487742029209996247990909, 9.555401770216505775026403185362, 10.22067562738484038150612112691

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