Properties

Label 2-40310-1.1-c1-0-24
Degree $2$
Conductor $40310$
Sign $-1$
Analytic cond. $321.876$
Root an. cond. $17.9409$
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  − 2-s + 2·3-s + 4-s + 5-s − 2·6-s + 3·7-s − 8-s + 9-s − 10-s + 3·11-s + 2·12-s − 3·14-s + 2·15-s + 16-s − 2·17-s − 18-s + 2·19-s + 20-s + 6·21-s − 3·22-s − 8·23-s − 2·24-s + 25-s − 4·27-s + 3·28-s − 29-s − 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.904·11-s + 0.577·12-s − 0.801·14-s + 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.458·19-s + 0.223·20-s + 1.30·21-s − 0.639·22-s − 1.66·23-s − 0.408·24-s + 1/5·25-s − 0.769·27-s + 0.566·28-s − 0.185·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40310\)    =    \(2 \cdot 5 \cdot 29 \cdot 139\)
Sign: $-1$
Analytic conductor: \(321.876\)
Root analytic conductor: \(17.9409\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 40310,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 - T \)
29 \( 1 + T \)
139 \( 1 - T \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−14.85446207913149, −14.57048833394879, −14.10294366681140, −13.65102216643120, −13.18226929132561, −12.28062788331929, −11.75742371402658, −11.45553807058251, −10.71945826038191, −10.17953475186831, −9.560709876810498, −9.196695301917563, −8.557964033533381, −8.304826037013706, −7.686048521374208, −7.227048370744293, −6.434308909710036, −5.924739777032021, −5.149610800520353, −4.452519661266333, −3.717890713633589, −3.176227044234058, −2.246941015005402, −1.831303841422675, −1.331168576801413, 0, 1.331168576801413, 1.831303841422675, 2.246941015005402, 3.176227044234058, 3.717890713633589, 4.452519661266333, 5.149610800520353, 5.924739777032021, 6.434308909710036, 7.227048370744293, 7.686048521374208, 8.304826037013706, 8.557964033533381, 9.196695301917563, 9.560709876810498, 10.17953475186831, 10.71945826038191, 11.45553807058251, 11.75742371402658, 12.28062788331929, 13.18226929132561, 13.65102216643120, 14.10294366681140, 14.57048833394879, 14.85446207913149

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