Properties

Label 2-304-1.1-c3-0-24
Degree $2$
Conductor $304$
Sign $-1$
Analytic cond. $17.9365$
Root an. cond. $4.23516$
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  + 5.61·3-s − 13.8·5-s + 9.22·7-s + 4.56·9-s − 59.8·11-s + 38.2·13-s − 77.6·15-s − 47.0·17-s + 19·19-s + 51.8·21-s − 179.·23-s + 66.1·25-s − 126.·27-s − 96.6·29-s − 125.·31-s − 336.·33-s − 127.·35-s + 407.·37-s + 214.·39-s − 220.·41-s − 100.·43-s − 63.0·45-s + 213.·47-s − 257.·49-s − 264.·51-s + 19.8·53-s + 827.·55-s + ⋯
L(s)  = 1  + 1.08·3-s − 1.23·5-s + 0.497·7-s + 0.169·9-s − 1.64·11-s + 0.815·13-s − 1.33·15-s − 0.671·17-s + 0.229·19-s + 0.538·21-s − 1.62·23-s + 0.529·25-s − 0.898·27-s − 0.618·29-s − 0.729·31-s − 1.77·33-s − 0.615·35-s + 1.81·37-s + 0.881·39-s − 0.840·41-s − 0.357·43-s − 0.209·45-s + 0.663·47-s − 0.752·49-s − 0.726·51-s + 0.0513·53-s + 2.02·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-1$
Analytic conductor: \(17.9365\)
Root analytic conductor: \(4.23516\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 304,\ (\ :3/2),\ -1)\)

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 \( 1 - 5.61T + 27T^{2} \)
5 \( 1 + 13.8T + 125T^{2} \)
7 \( 1 - 9.22T + 343T^{2} \)
11 \( 1 + 59.8T + 1.33e3T^{2} \)
13 \( 1 - 38.2T + 2.19e3T^{2} \)
17 \( 1 + 47.0T + 4.91e3T^{2} \)
23 \( 1 + 179.T + 1.21e4T^{2} \)
29 \( 1 + 96.6T + 2.43e4T^{2} \)
31 \( 1 + 125.T + 2.97e4T^{2} \)
37 \( 1 - 407.T + 5.06e4T^{2} \)
41 \( 1 + 220.T + 6.89e4T^{2} \)
43 \( 1 + 100.T + 7.95e4T^{2} \)
47 \( 1 - 213.T + 1.03e5T^{2} \)
53 \( 1 - 19.8T + 1.48e5T^{2} \)
59 \( 1 - 97.7T + 2.05e5T^{2} \)
61 \( 1 - 266.T + 2.26e5T^{2} \)
67 \( 1 + 627.T + 3.00e5T^{2} \)
71 \( 1 - 644.T + 3.57e5T^{2} \)
73 \( 1 - 807.T + 3.89e5T^{2} \)
79 \( 1 + 785.T + 4.93e5T^{2} \)
83 \( 1 + 1.17e3T + 5.71e5T^{2} \)
89 \( 1 + 234.T + 7.04e5T^{2} \)
97 \( 1 + 1.34e3T + 9.12e5T^{2} \)
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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

−10.99034085167293254996170689133, −9.795715569884797775139934354644, −8.507810117694480635837482060761, −8.079308047283427314572612299458, −7.41846989957823225875675103310, −5.73667459731354845032724034809, −4.35366855168567833371880230420, −3.39450308257032131912545305695, −2.18539184943696806942614263097, 0, 2.18539184943696806942614263097, 3.39450308257032131912545305695, 4.35366855168567833371880230420, 5.73667459731354845032724034809, 7.41846989957823225875675103310, 8.079308047283427314572612299458, 8.507810117694480635837482060761, 9.795715569884797775139934354644, 10.99034085167293254996170689133

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