Properties

Label 2-19320-1.1-c1-0-9
Degree $2$
Conductor $19320$
Sign $1$
Analytic cond. $154.270$
Root an. cond. $12.4205$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 7-s + 9-s − 4·11-s + 2·13-s + 15-s − 2·17-s − 4·19-s + 21-s − 23-s + 25-s + 27-s + 2·29-s − 4·33-s + 35-s + 6·37-s + 2·39-s − 6·41-s − 8·43-s + 45-s + 4·47-s + 49-s − 2·51-s + 2·53-s − 4·55-s − 4·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s − 0.485·17-s − 0.917·19-s + 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.696·33-s + 0.169·35-s + 0.986·37-s + 0.320·39-s − 0.937·41-s − 1.21·43-s + 0.149·45-s + 0.583·47-s + 1/7·49-s − 0.280·51-s + 0.274·53-s − 0.539·55-s − 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(154.270\)
Root analytic conductor: \(12.4205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{19320} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 19320,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.760618118\)
\(L(\frac12)\) \(\approx\) \(2.760618118\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
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 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−15.47825728579982, −15.23563072150296, −14.67975317896505, −13.92852467323407, −13.57822503596992, −13.05802066288190, −12.62815086964912, −11.87849598712204, −11.07963571984886, −10.77367575185495, −10.02949771799974, −9.693338552739712, −8.770805996226832, −8.353106984664251, −8.013346032978059, −7.108731621069041, −6.608418800465490, −5.844594597081920, −5.207954206875644, −4.563310565655678, −3.869972813330071, −3.034828380757249, −2.313633748715203, −1.810779412154449, −0.6505998521477266, 0.6505998521477266, 1.810779412154449, 2.313633748715203, 3.034828380757249, 3.869972813330071, 4.563310565655678, 5.207954206875644, 5.844594597081920, 6.608418800465490, 7.108731621069041, 8.013346032978059, 8.353106984664251, 8.770805996226832, 9.693338552739712, 10.02949771799974, 10.77367575185495, 11.07963571984886, 11.87849598712204, 12.62815086964912, 13.05802066288190, 13.57822503596992, 13.92852467323407, 14.67975317896505, 15.23563072150296, 15.47825728579982

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