Properties

Label 2-608-1.1-c3-0-48
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.55·3-s + 7.40·5-s + 10.4·7-s − 20.4·9-s − 48.7·11-s + 4.45·13-s + 18.8·15-s − 61.5·17-s − 19·19-s + 26.7·21-s − 34.8·23-s − 70.2·25-s − 121.·27-s − 179.·29-s − 132.·31-s − 124.·33-s + 77.6·35-s − 99.3·37-s + 11.3·39-s + 382.·41-s + 129.·43-s − 151.·45-s + 488.·47-s − 232.·49-s − 157.·51-s − 129.·53-s − 361.·55-s + ⋯
L(s)  = 1  + 0.491·3-s + 0.661·5-s + 0.566·7-s − 0.758·9-s − 1.33·11-s + 0.0950·13-s + 0.325·15-s − 0.877·17-s − 0.229·19-s + 0.278·21-s − 0.315·23-s − 0.561·25-s − 0.864·27-s − 1.14·29-s − 0.769·31-s − 0.657·33-s + 0.375·35-s − 0.441·37-s + 0.0466·39-s + 1.45·41-s + 0.459·43-s − 0.502·45-s + 1.51·47-s − 0.678·49-s − 0.431·51-s − 0.334·53-s − 0.885·55-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: \(1\)
Selberg data: \((2,\ 608,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.55T + 27T^{2} \)
5 \( 1 - 7.40T + 125T^{2} \)
7 \( 1 - 10.4T + 343T^{2} \)
11 \( 1 + 48.7T + 1.33e3T^{2} \)
13 \( 1 - 4.45T + 2.19e3T^{2} \)
17 \( 1 + 61.5T + 4.91e3T^{2} \)
23 \( 1 + 34.8T + 1.21e4T^{2} \)
29 \( 1 + 179.T + 2.43e4T^{2} \)
31 \( 1 + 132.T + 2.97e4T^{2} \)
37 \( 1 + 99.3T + 5.06e4T^{2} \)
41 \( 1 - 382.T + 6.89e4T^{2} \)
43 \( 1 - 129.T + 7.95e4T^{2} \)
47 \( 1 - 488.T + 1.03e5T^{2} \)
53 \( 1 + 129.T + 1.48e5T^{2} \)
59 \( 1 - 479.T + 2.05e5T^{2} \)
61 \( 1 + 394.T + 2.26e5T^{2} \)
67 \( 1 - 19.8T + 3.00e5T^{2} \)
71 \( 1 + 221.T + 3.57e5T^{2} \)
73 \( 1 - 258.T + 3.89e5T^{2} \)
79 \( 1 + 469.T + 4.93e5T^{2} \)
83 \( 1 - 373.T + 5.71e5T^{2} \)
89 \( 1 + 1.60e3T + 7.04e5T^{2} \)
97 \( 1 + 1.65e3T + 9.12e5T^{2} \)
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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

−9.695103116629293343123989677724, −8.914525800712027624980967856145, −8.098109042580773256733744512775, −7.33824600170806685008366686941, −5.92878137447091350867231736083, −5.35446245632999007283173180894, −4.07871224843389709113405789698, −2.68762737406417200615796665213, −1.94333165002042830225112217241, 0, 1.94333165002042830225112217241, 2.68762737406417200615796665213, 4.07871224843389709113405789698, 5.35446245632999007283173180894, 5.92878137447091350867231736083, 7.33824600170806685008366686941, 8.098109042580773256733744512775, 8.914525800712027624980967856145, 9.695103116629293343123989677724

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