Properties

Label 2-247962-1.1-c1-0-14
Degree $2$
Conductor $247962$
Sign $-1$
Analytic cond. $1979.98$
Root an. cond. $44.4970$
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 − 3-s + 4-s − 4·5-s + 6-s − 8-s + 9-s + 4·10-s − 11-s − 12-s − 13-s + 4·15-s + 16-s − 18-s + 2·19-s − 4·20-s + 22-s + 4·23-s + 24-s + 11·25-s + 26-s − 27-s − 2·29-s − 4·30-s − 32-s + 33-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.301·11-s − 0.288·12-s − 0.277·13-s + 1.03·15-s + 1/4·16-s − 0.235·18-s + 0.458·19-s − 0.894·20-s + 0.213·22-s + 0.834·23-s + 0.204·24-s + 11/5·25-s + 0.196·26-s − 0.192·27-s − 0.371·29-s − 0.730·30-s − 0.176·32-s + 0.174·33-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 247962 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247962 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(247962\)    =    \(2 \cdot 3 \cdot 11 \cdot 13 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(1979.98\)
Root analytic conductor: \(44.4970\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 247962,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 \)
good5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−12.84971479768075, −12.63418977794745, −11.86931941894764, −11.53492421452886, −11.41305947208371, −10.92795448758438, −10.32877022637105, −9.869509788896918, −9.470277991975822, −8.691288054760425, −8.371836978342357, −7.954245805182781, −7.455018001730324, −7.078332615183113, −6.642572051041170, −6.118054315790410, −5.247152047963715, −4.925601628458840, −4.483681591814568, −3.661875409319856, −3.350679303246516, −2.816018433470652, −1.928344441093294, −1.223581169231357, −0.5431731298781314, 0, 0.5431731298781314, 1.223581169231357, 1.928344441093294, 2.816018433470652, 3.350679303246516, 3.661875409319856, 4.483681591814568, 4.925601628458840, 5.247152047963715, 6.118054315790410, 6.642572051041170, 7.078332615183113, 7.455018001730324, 7.954245805182781, 8.371836978342357, 8.691288054760425, 9.470277991975822, 9.869509788896918, 10.32877022637105, 10.92795448758438, 11.41305947208371, 11.53492421452886, 11.86931941894764, 12.63418977794745, 12.84971479768075

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