Properties

Label 2-608-1.1-c3-0-46
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  + 7.68·3-s − 15.2·5-s − 2.79·7-s + 32.0·9-s − 0.456·11-s − 0.618·13-s − 117.·15-s + 12.2·17-s − 19·19-s − 21.4·21-s − 116.·23-s + 107.·25-s + 38.9·27-s + 100.·29-s − 206.·31-s − 3.50·33-s + 42.6·35-s − 212.·37-s − 4.75·39-s − 42.1·41-s − 34.2·43-s − 489.·45-s − 393.·47-s − 335.·49-s + 94.1·51-s − 497.·53-s + 6.96·55-s + ⋯
L(s)  = 1  + 1.47·3-s − 1.36·5-s − 0.150·7-s + 1.18·9-s − 0.0125·11-s − 0.0131·13-s − 2.01·15-s + 0.174·17-s − 0.229·19-s − 0.223·21-s − 1.05·23-s + 0.863·25-s + 0.277·27-s + 0.644·29-s − 1.19·31-s − 0.0184·33-s + 0.205·35-s − 0.945·37-s − 0.0195·39-s − 0.160·41-s − 0.121·43-s − 1.62·45-s − 1.22·47-s − 0.977·49-s + 0.258·51-s − 1.28·53-s + 0.0170·55-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 \( 1 - 7.68T + 27T^{2} \)
5 \( 1 + 15.2T + 125T^{2} \)
7 \( 1 + 2.79T + 343T^{2} \)
11 \( 1 + 0.456T + 1.33e3T^{2} \)
13 \( 1 + 0.618T + 2.19e3T^{2} \)
17 \( 1 - 12.2T + 4.91e3T^{2} \)
23 \( 1 + 116.T + 1.21e4T^{2} \)
29 \( 1 - 100.T + 2.43e4T^{2} \)
31 \( 1 + 206.T + 2.97e4T^{2} \)
37 \( 1 + 212.T + 5.06e4T^{2} \)
41 \( 1 + 42.1T + 6.89e4T^{2} \)
43 \( 1 + 34.2T + 7.95e4T^{2} \)
47 \( 1 + 393.T + 1.03e5T^{2} \)
53 \( 1 + 497.T + 1.48e5T^{2} \)
59 \( 1 + 82.9T + 2.05e5T^{2} \)
61 \( 1 + 655.T + 2.26e5T^{2} \)
67 \( 1 + 88.1T + 3.00e5T^{2} \)
71 \( 1 - 430.T + 3.57e5T^{2} \)
73 \( 1 - 65.8T + 3.89e5T^{2} \)
79 \( 1 - 528.T + 4.93e5T^{2} \)
83 \( 1 - 592.T + 5.71e5T^{2} \)
89 \( 1 - 1.15e3T + 7.04e5T^{2} \)
97 \( 1 + 51.7T + 9.12e5T^{2} \)
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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

−9.590643220958454715465078550443, −8.769073670008496859948265717694, −7.989475729882922518766621237017, −7.58734375108968502431442303635, −6.43864115170584110685367840201, −4.80688935017697010948008845773, −3.74659313720450847760137994827, −3.21353255774194248801428303076, −1.86545654666441810485161251732, 0, 1.86545654666441810485161251732, 3.21353255774194248801428303076, 3.74659313720450847760137994827, 4.80688935017697010948008845773, 6.43864115170584110685367840201, 7.58734375108968502431442303635, 7.989475729882922518766621237017, 8.769073670008496859948265717694, 9.590643220958454715465078550443

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