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Properties

Label 2-223440-1.1-c1-0-6
Degree $2$
Conductor $223440$
Sign $1$
Analytic cond. $1784.17$
Root an. cond. $42.2395$
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 + 9-s + 3·11-s − 7·13-s − 15-s − 6·17-s − 19-s − 7·23-s + 25-s − 27-s − 31-s − 3·33-s + 2·37-s + 7·39-s + 6·41-s − 9·43-s + 45-s + 6·47-s + 6·51-s + 9·53-s + 3·55-s + 57-s + 9·59-s + 11·61-s − 7·65-s − 6·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.904·11-s − 1.94·13-s − 0.258·15-s − 1.45·17-s − 0.229·19-s − 1.45·23-s + 1/5·25-s − 0.192·27-s − 0.179·31-s − 0.522·33-s + 0.328·37-s + 1.12·39-s + 0.937·41-s − 1.37·43-s + 0.149·45-s + 0.875·47-s + 0.840·51-s + 1.23·53-s + 0.404·55-s + 0.132·57-s + 1.17·59-s + 1.40·61-s − 0.868·65-s − 0.733·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 223440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 223440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(223440\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(1784.17\)
Root analytic conductor: \(42.2395\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 223440,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6015549483\)
\(L(\frac12)\) \(\approx\) \(0.6015549483\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + T \)
good11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 5 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−13.04929705827945, −12.24774122370476, −12.05724899145420, −11.72284300107297, −11.12010539629166, −10.59855992425215, −10.12600876705243, −9.694714071409125, −9.397714108067285, −8.704241294816174, −8.384564238759172, −7.515760186575017, −7.224539147241282, −6.675329754038796, −6.339644214661031, −5.653897058608112, −5.332690398704972, −4.580508926589451, −4.270040326635045, −3.826685846271792, −2.824514569317866, −2.282811453280535, −1.957214963553001, −1.126507255367658, −0.2291441115094806, 0.2291441115094806, 1.126507255367658, 1.957214963553001, 2.282811453280535, 2.824514569317866, 3.826685846271792, 4.270040326635045, 4.580508926589451, 5.332690398704972, 5.653897058608112, 6.339644214661031, 6.675329754038796, 7.224539147241282, 7.515760186575017, 8.384564238759172, 8.704241294816174, 9.397714108067285, 9.694714071409125, 10.12600876705243, 10.59855992425215, 11.12010539629166, 11.72284300107297, 12.05724899145420, 12.24774122370476, 13.04929705827945

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