Properties

Label 2-388416-1.1-c1-0-157
Degree $2$
Conductor $388416$
Sign $-1$
Analytic cond. $3101.51$
Root an. cond. $55.6912$
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 − 3·5-s − 7-s + 9-s + 5·11-s + 3·15-s + 6·19-s + 21-s + 2·23-s + 4·25-s − 27-s + 9·29-s + 3·31-s − 5·33-s + 3·35-s − 6·37-s + 6·41-s − 4·43-s − 3·45-s − 6·47-s + 49-s − 3·53-s − 15·55-s − 6·57-s − 13·59-s − 4·61-s − 63-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s + 1.50·11-s + 0.774·15-s + 1.37·19-s + 0.218·21-s + 0.417·23-s + 4/5·25-s − 0.192·27-s + 1.67·29-s + 0.538·31-s − 0.870·33-s + 0.507·35-s − 0.986·37-s + 0.937·41-s − 0.609·43-s − 0.447·45-s − 0.875·47-s + 1/7·49-s − 0.412·53-s − 2.02·55-s − 0.794·57-s − 1.69·59-s − 0.512·61-s − 0.125·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(388416\)    =    \(2^{6} \cdot 3 \cdot 7 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(3101.51\)
Root analytic conductor: \(55.6912\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{388416} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 388416,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 + T \)
17 \( 1 \)
good5 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−12.32118495572906, −12.21309601069766, −11.79023230361924, −11.47910467213251, −11.02787757376562, −10.54712459809986, −9.854449595999468, −9.686906044520046, −8.969673915217783, −8.647726990666593, −8.124186126152155, −7.421923432159422, −7.335360109988932, −6.637200437853013, −6.323615722796245, −5.880397840316778, −4.988294169946996, −4.766861982983373, −4.234260212101295, −3.675283526131096, −3.239472857588195, −2.882995795883699, −1.818394976030829, −1.162299383058667, −0.7722083053086834, 0, 0.7722083053086834, 1.162299383058667, 1.818394976030829, 2.882995795883699, 3.239472857588195, 3.675283526131096, 4.234260212101295, 4.766861982983373, 4.988294169946996, 5.880397840316778, 6.323615722796245, 6.637200437853013, 7.335360109988932, 7.421923432159422, 8.124186126152155, 8.647726990666593, 8.969673915217783, 9.686906044520046, 9.854449595999468, 10.54712459809986, 11.02787757376562, 11.47910467213251, 11.79023230361924, 12.21309601069766, 12.32118495572906

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