Properties

Label 2-123840-1.1-c1-0-122
Degree $2$
Conductor $123840$
Sign $-1$
Analytic cond. $988.867$
Root an. cond. $31.4462$
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  − 5-s − 2·7-s + 3·11-s − 13-s + 5·17-s + 6·19-s − 23-s + 25-s − 2·29-s + 11·31-s + 2·35-s − 2·37-s + 3·41-s + 43-s − 12·47-s − 3·49-s + 9·53-s − 3·55-s + 12·61-s + 65-s − 15·67-s + 12·71-s − 4·73-s − 6·77-s − 8·79-s − 3·83-s − 5·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s + 0.904·11-s − 0.277·13-s + 1.21·17-s + 1.37·19-s − 0.208·23-s + 1/5·25-s − 0.371·29-s + 1.97·31-s + 0.338·35-s − 0.328·37-s + 0.468·41-s + 0.152·43-s − 1.75·47-s − 3/7·49-s + 1.23·53-s − 0.404·55-s + 1.53·61-s + 0.124·65-s − 1.83·67-s + 1.42·71-s − 0.468·73-s − 0.683·77-s − 0.900·79-s − 0.329·83-s − 0.542·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(123840\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 43\)
Sign: $-1$
Analytic conductor: \(988.867\)
Root analytic conductor: \(31.4462\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 123840,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
43 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 11 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 15 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−13.81707083192784, −13.25273099231686, −12.83380364521864, −12.12305106268478, −11.89950326179281, −11.58626135897282, −10.95472135516957, −10.10875833914766, −9.891963398817001, −9.601557953431061, −8.831905209159050, −8.442669299793664, −7.742421658655829, −7.404873488493944, −6.853207000487574, −6.236233085208581, −5.918385390805973, −5.069882411609143, −4.774290134100806, −3.821752331796604, −3.589419686407125, −2.981808016618459, −2.404377927454904, −1.313600263756290, −0.9710693309374795, 0, 0.9710693309374795, 1.313600263756290, 2.404377927454904, 2.981808016618459, 3.589419686407125, 3.821752331796604, 4.774290134100806, 5.069882411609143, 5.918385390805973, 6.236233085208581, 6.853207000487574, 7.404873488493944, 7.742421658655829, 8.442669299793664, 8.831905209159050, 9.601557953431061, 9.891963398817001, 10.10875833914766, 10.95472135516957, 11.58626135897282, 11.89950326179281, 12.12305106268478, 12.83380364521864, 13.25273099231686, 13.81707083192784

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