Properties

Label 2-10-1.1-c21-0-6
Degree $2$
Conductor $10$
Sign $-1$
Analytic cond. $27.9477$
Root an. cond. $5.28656$
Motivic weight $21$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.02e3·2-s − 2.19e4·3-s + 1.04e6·4-s + 9.76e6·5-s − 2.24e7·6-s − 7.22e8·7-s + 1.07e9·8-s − 9.97e9·9-s + 1.00e10·10-s + 4.99e10·11-s − 2.29e10·12-s − 2.43e10·13-s − 7.40e11·14-s − 2.14e11·15-s + 1.09e12·16-s − 5.76e12·17-s − 1.02e13·18-s − 3.02e13·19-s + 1.02e13·20-s + 1.58e13·21-s + 5.11e13·22-s − 1.45e14·23-s − 2.35e13·24-s + 9.53e13·25-s − 2.49e13·26-s + 4.48e14·27-s − 7.57e14·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.214·3-s + 1/2·4-s + 0.447·5-s − 0.151·6-s − 0.967·7-s + 0.353·8-s − 0.954·9-s + 0.316·10-s + 0.580·11-s − 0.107·12-s − 0.0489·13-s − 0.683·14-s − 0.0958·15-s + 1/4·16-s − 0.694·17-s − 0.674·18-s − 1.13·19-s + 0.223·20-s + 0.207·21-s + 0.410·22-s − 0.732·23-s − 0.0757·24-s + 1/5·25-s − 0.0346·26-s + 0.418·27-s − 0.483·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-1$
Analytic conductor: \(27.9477\)
Root analytic conductor: \(5.28656\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 10,\ (\ :21/2),\ -1)\)

Particular Values

\(L(11)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{10} T \)
5 \( 1 - p^{10} T \)
good3 \( 1 + 812 p^{3} T + p^{21} T^{2} \)
7 \( 1 + 103250464 p T + p^{21} T^{2} \)
11 \( 1 - 49976398572 T + p^{21} T^{2} \)
13 \( 1 + 24351400354 T + p^{21} T^{2} \)
17 \( 1 + 5768874283278 T + p^{21} T^{2} \)
19 \( 1 + 1592117156980 p T + p^{21} T^{2} \)
23 \( 1 + 145464074718144 T + p^{21} T^{2} \)
29 \( 1 + 1167107530943250 T + p^{21} T^{2} \)
31 \( 1 + 7431907384909648 T + p^{21} T^{2} \)
37 \( 1 + 54621676523378698 T + p^{21} T^{2} \)
41 \( 1 + 20889593910177078 T + p^{21} T^{2} \)
43 \( 1 - 76193896658948996 T + p^{21} T^{2} \)
47 \( 1 - 508989722403999912 T + p^{21} T^{2} \)
53 \( 1 - 1341490659556639206 T + p^{21} T^{2} \)
59 \( 1 - 2306403035584927500 T + p^{21} T^{2} \)
61 \( 1 - 5268341017122878702 T + p^{21} T^{2} \)
67 \( 1 + 10904975405120900788 T + p^{21} T^{2} \)
71 \( 1 + 13367810994741254088 T + p^{21} T^{2} \)
73 \( 1 + 17952853611107729014 T + p^{21} T^{2} \)
79 \( 1 - \)\(11\!\cdots\!40\)\( T + p^{21} T^{2} \)
83 \( 1 - \)\(13\!\cdots\!16\)\( T + p^{21} T^{2} \)
89 \( 1 - \)\(34\!\cdots\!90\)\( T + p^{21} T^{2} \)
97 \( 1 + \)\(37\!\cdots\!98\)\( T + p^{21} T^{2} \)
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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

−14.75353070782327810408187989146, −13.48255903945352443555985243566, −12.21517119658238140377038067924, −10.70268993355063226246363591174, −8.979595567902947804469868499217, −6.71509026840207069580152100218, −5.62516187008915379714580773687, −3.75276667295138783384212209750, −2.19543771843557833097027154008, 0, 2.19543771843557833097027154008, 3.75276667295138783384212209750, 5.62516187008915379714580773687, 6.71509026840207069580152100218, 8.979595567902947804469868499217, 10.70268993355063226246363591174, 12.21517119658238140377038067924, 13.48255903945352443555985243566, 14.75353070782327810408187989146

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