Properties

Label 2-608-1.1-c3-0-37
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  − 5.31·3-s + 9.41·5-s + 16.2·7-s + 1.29·9-s + 8.71·11-s − 43.4·13-s − 50.0·15-s − 124.·17-s − 19·19-s − 86.4·21-s + 71.6·23-s − 36.3·25-s + 136.·27-s + 279.·29-s + 252.·31-s − 46.3·33-s + 152.·35-s + 17.9·37-s + 230.·39-s − 466.·41-s − 340.·43-s + 12.2·45-s − 305.·47-s − 78.9·49-s + 661.·51-s − 293.·53-s + 82.0·55-s + ⋯
L(s)  = 1  − 1.02·3-s + 0.841·5-s + 0.877·7-s + 0.0480·9-s + 0.238·11-s − 0.926·13-s − 0.861·15-s − 1.77·17-s − 0.229·19-s − 0.898·21-s + 0.649·23-s − 0.291·25-s + 0.974·27-s + 1.79·29-s + 1.46·31-s − 0.244·33-s + 0.738·35-s + 0.0797·37-s + 0.948·39-s − 1.77·41-s − 1.20·43-s + 0.0404·45-s − 0.947·47-s − 0.230·49-s + 1.81·51-s − 0.760·53-s + 0.201·55-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 \( 1 + 5.31T + 27T^{2} \)
5 \( 1 - 9.41T + 125T^{2} \)
7 \( 1 - 16.2T + 343T^{2} \)
11 \( 1 - 8.71T + 1.33e3T^{2} \)
13 \( 1 + 43.4T + 2.19e3T^{2} \)
17 \( 1 + 124.T + 4.91e3T^{2} \)
23 \( 1 - 71.6T + 1.21e4T^{2} \)
29 \( 1 - 279.T + 2.43e4T^{2} \)
31 \( 1 - 252.T + 2.97e4T^{2} \)
37 \( 1 - 17.9T + 5.06e4T^{2} \)
41 \( 1 + 466.T + 6.89e4T^{2} \)
43 \( 1 + 340.T + 7.95e4T^{2} \)
47 \( 1 + 305.T + 1.03e5T^{2} \)
53 \( 1 + 293.T + 1.48e5T^{2} \)
59 \( 1 + 408.T + 2.05e5T^{2} \)
61 \( 1 - 334.T + 2.26e5T^{2} \)
67 \( 1 + 754.T + 3.00e5T^{2} \)
71 \( 1 + 758.T + 3.57e5T^{2} \)
73 \( 1 + 393.T + 3.89e5T^{2} \)
79 \( 1 - 91.7T + 4.93e5T^{2} \)
83 \( 1 - 1.39e3T + 5.71e5T^{2} \)
89 \( 1 - 931.T + 7.04e5T^{2} \)
97 \( 1 + 282.T + 9.12e5T^{2} \)
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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

−10.05441611134414179603575075128, −8.939268840132889777287675867274, −8.134833816666158436048106905002, −6.68873381952675835553215304297, −6.31625169843796775730669192301, −4.92313746841813392491061486715, −4.75429356538495847910011215379, −2.70607830682827142597786644160, −1.53530547015177849487995772050, 0, 1.53530547015177849487995772050, 2.70607830682827142597786644160, 4.75429356538495847910011215379, 4.92313746841813392491061486715, 6.31625169843796775730669192301, 6.68873381952675835553215304297, 8.134833816666158436048106905002, 8.939268840132889777287675867274, 10.05441611134414179603575075128

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