Properties

Label 2-19074-1.1-c1-0-32
Degree $2$
Conductor $19074$
Sign $-1$
Analytic cond. $152.306$
Root an. cond. $12.3412$
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  + 2-s − 3-s + 4-s + 2·5-s − 6-s + 4·7-s + 8-s + 9-s + 2·10-s + 11-s − 12-s − 4·13-s + 4·14-s − 2·15-s + 16-s + 18-s − 8·19-s + 2·20-s − 4·21-s + 22-s − 24-s − 25-s − 4·26-s − 27-s + 4·28-s − 2·30-s − 10·31-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.288·12-s − 1.10·13-s + 1.06·14-s − 0.516·15-s + 1/4·16-s + 0.235·18-s − 1.83·19-s + 0.447·20-s − 0.872·21-s + 0.213·22-s − 0.204·24-s − 1/5·25-s − 0.784·26-s − 0.192·27-s + 0.755·28-s − 0.365·30-s − 1.79·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19074 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19074 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19074\)    =    \(2 \cdot 3 \cdot 11 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(152.306\)
Root analytic conductor: \(12.3412\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 19074,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 + T \)
11 \( 1 - T \)
17 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
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

−15.92435305180826, −15.25953559903750, −14.73540044173146, −14.38441916270977, −13.97610673142036, −13.23128001090247, −12.55619582781955, −12.36539461376133, −11.53720959110764, −11.07587029269359, −10.61734158458755, −10.08410885720174, −9.274363714260776, −8.724533450183503, −7.901144872405312, −7.353957884656157, −6.724898861698675, −6.002285438101875, −5.520110347795975, −4.923388887671205, −4.442883185632045, −3.784805151837642, −2.562103014599905, −1.913484352984504, −1.523914228386670, 0, 1.523914228386670, 1.913484352984504, 2.562103014599905, 3.784805151837642, 4.442883185632045, 4.923388887671205, 5.520110347795975, 6.002285438101875, 6.724898861698675, 7.353957884656157, 7.901144872405312, 8.724533450183503, 9.274363714260776, 10.08410885720174, 10.61734158458755, 11.07587029269359, 11.53720959110764, 12.36539461376133, 12.55619582781955, 13.23128001090247, 13.97610673142036, 14.38441916270977, 14.73540044173146, 15.25953559903750, 15.92435305180826

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