Properties

Label 2-215475-1.1-c1-0-19
Degree $2$
Conductor $215475$
Sign $1$
Analytic cond. $1720.57$
Root an. cond. $41.4798$
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  − 2·2-s − 3-s + 2·4-s + 2·6-s + 2·7-s + 9-s − 2·12-s − 4·14-s − 4·16-s − 17-s − 2·18-s + 8·19-s − 2·21-s + 6·23-s − 27-s + 4·28-s + 3·29-s − 11·31-s + 8·32-s + 2·34-s + 2·36-s − 6·37-s − 16·38-s + 4·42-s − 12·46-s − 2·47-s + 4·48-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s + 0.755·7-s + 1/3·9-s − 0.577·12-s − 1.06·14-s − 16-s − 0.242·17-s − 0.471·18-s + 1.83·19-s − 0.436·21-s + 1.25·23-s − 0.192·27-s + 0.755·28-s + 0.557·29-s − 1.97·31-s + 1.41·32-s + 0.342·34-s + 1/3·36-s − 0.986·37-s − 2.59·38-s + 0.617·42-s − 1.76·46-s − 0.291·47-s + 0.577·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 215475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(215475\)    =    \(3 \cdot 5^{2} \cdot 13^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(1720.57\)
Root analytic conductor: \(41.4798\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 215475,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.354697800\)
\(L(\frac12)\) \(\approx\) \(1.354697800\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 \)
13 \( 1 \)
17 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 11 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−12.96895173505937, −12.31414653885548, −11.84588680377580, −11.38324044919957, −11.07523345227071, −10.62337886230655, −10.22790871222339, −9.618784604497219, −9.152073108712423, −8.968655745139370, −8.218308490381263, −7.840277171739862, −7.349264726065248, −6.955623382081368, −6.572025730612780, −5.675271735793247, −5.167376368857252, −4.979413018821758, −4.213723110285109, −3.491384272695407, −2.949855568202891, −1.954253996116994, −1.721316375657183, −0.8773279770723375, −0.5710481256047217, 0.5710481256047217, 0.8773279770723375, 1.721316375657183, 1.954253996116994, 2.949855568202891, 3.491384272695407, 4.213723110285109, 4.979413018821758, 5.167376368857252, 5.675271735793247, 6.572025730612780, 6.955623382081368, 7.349264726065248, 7.840277171739862, 8.218308490381263, 8.968655745139370, 9.152073108712423, 9.618784604497219, 10.22790871222339, 10.62337886230655, 11.07523345227071, 11.38324044919957, 11.84588680377580, 12.31414653885548, 12.96895173505937

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