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Properties

Label 2-17328-1.1-c1-0-17
Degree $2$
Conductor $17328$
Sign $-1$
Analytic cond. $138.364$
Root an. cond. $11.7628$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·5-s + 5·7-s + 9-s − 11-s − 2·13-s + 3·15-s − 17-s − 5·21-s + 4·23-s + 4·25-s − 27-s + 2·29-s − 6·31-s + 33-s − 15·35-s + 2·39-s + 43-s − 3·45-s + 9·47-s + 18·49-s + 51-s − 10·53-s + 3·55-s − 8·59-s − 61-s + 5·63-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s + 1.88·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.774·15-s − 0.242·17-s − 1.09·21-s + 0.834·23-s + 4/5·25-s − 0.192·27-s + 0.371·29-s − 1.07·31-s + 0.174·33-s − 2.53·35-s + 0.320·39-s + 0.152·43-s − 0.447·45-s + 1.31·47-s + 18/7·49-s + 0.140·51-s − 1.37·53-s + 0.404·55-s − 1.04·59-s − 0.128·61-s + 0.629·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17328\)    =    \(2^{4} \cdot 3 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(138.364\)
Root analytic conductor: \(11.7628\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 17328,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
19 \( 1 \)
good5 \( 1 + 3 T + p T^{2} \)
7 \( 1 - 5 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−15.95514422714751, −15.62983248135334, −15.00130715232562, −14.65367693657247, −14.09552876345338, −13.34501591363017, −12.54407895363717, −12.16434451522927, −11.59492874915826, −11.14351619276312, −10.83558593183010, −10.21291942930301, −9.158278110827002, −8.685187497794598, −7.978860804577929, −7.489194641341449, −7.270791516439911, −6.254992161798117, −5.366006079185088, −4.893907391599638, −4.428086174341133, −3.798243340943585, −2.802508831646139, −1.866423160967098, −1.026025901219009, 0, 1.026025901219009, 1.866423160967098, 2.802508831646139, 3.798243340943585, 4.428086174341133, 4.893907391599638, 5.366006079185088, 6.254992161798117, 7.270791516439911, 7.489194641341449, 7.978860804577929, 8.685187497794598, 9.158278110827002, 10.21291942930301, 10.83558593183010, 11.14351619276312, 11.59492874915826, 12.16434451522927, 12.54407895363717, 13.34501591363017, 14.09552876345338, 14.65367693657247, 15.00130715232562, 15.62983248135334, 15.95514422714751

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