Properties

Label 2-193200-1.1-c1-0-182
Degree $2$
Conductor $193200$
Sign $-1$
Analytic cond. $1542.70$
Root an. cond. $39.2773$
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 + 7-s + 9-s + 4·11-s + 2·13-s − 2·17-s + 4·19-s + 21-s − 23-s + 27-s − 2·29-s − 8·31-s + 4·33-s − 6·37-s + 2·39-s − 6·41-s − 4·43-s + 8·47-s + 49-s − 2·51-s + 10·53-s + 4·57-s + 12·59-s + 14·61-s + 63-s − 12·67-s − 69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s + 0.218·21-s − 0.208·23-s + 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.696·33-s − 0.986·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.280·51-s + 1.37·53-s + 0.529·57-s + 1.56·59-s + 1.79·61-s + 0.125·63-s − 1.46·67-s − 0.120·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(193200\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(1542.70\)
Root analytic conductor: \(39.2773\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{193200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 193200,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
7 \( 1 - T \)
23 \( 1 + T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.41752808275616, −13.00001025820994, −12.32245976276935, −11.88331524921587, −11.48935025319080, −11.09609877539083, −10.43432828051158, −9.994060538029345, −9.509549780813350, −8.891493655978804, −8.666644672653765, −8.307601021921914, −7.436773426026258, −7.089176391406803, −6.835034041940516, −5.970742291933274, −5.565059553805543, −5.043629866308999, −4.286406957258810, −3.758714049357601, −3.594197145752270, −2.743724370236125, −2.091186197020846, −1.511446463205115, −1.049450612167910, 0, 1.049450612167910, 1.511446463205115, 2.091186197020846, 2.743724370236125, 3.594197145752270, 3.758714049357601, 4.286406957258810, 5.043629866308999, 5.565059553805543, 5.970742291933274, 6.835034041940516, 7.089176391406803, 7.436773426026258, 8.307601021921914, 8.666644672653765, 8.891493655978804, 9.509549780813350, 9.994060538029345, 10.43432828051158, 11.09609877539083, 11.48935025319080, 11.88331524921587, 12.32245976276935, 13.00001025820994, 13.41752808275616

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