Properties

Label 2-97461-1.1-c1-0-5
Degree $2$
Conductor $97461$
Sign $1$
Analytic cond. $778.230$
Root an. cond. $27.8967$
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 + 2·4-s − 3·5-s + 6·10-s − 2·11-s − 13-s − 4·16-s + 17-s − 19-s − 6·20-s + 4·22-s − 3·23-s + 4·25-s + 2·26-s − 7·29-s + 5·31-s + 8·32-s − 2·34-s + 6·37-s + 2·38-s + 6·41-s + 11·43-s − 4·44-s + 6·46-s + 5·47-s − 8·50-s − 2·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.34·5-s + 1.89·10-s − 0.603·11-s − 0.277·13-s − 16-s + 0.242·17-s − 0.229·19-s − 1.34·20-s + 0.852·22-s − 0.625·23-s + 4/5·25-s + 0.392·26-s − 1.29·29-s + 0.898·31-s + 1.41·32-s − 0.342·34-s + 0.986·37-s + 0.324·38-s + 0.937·41-s + 1.67·43-s − 0.603·44-s + 0.884·46-s + 0.729·47-s − 1.13·50-s − 0.277·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(97461\)    =    \(3^{2} \cdot 7^{2} \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(778.230\)
Root analytic conductor: \(27.8967\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{97461} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 97461,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5672695153\)
\(L(\frac12)\) \(\approx\) \(0.5672695153\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 + T \)
17 \( 1 - T \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 - 15 T + p T^{2} \)
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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

−13.87404577220165, −13.11834742812820, −12.73794348947713, −12.12217926716007, −11.67898875314258, −11.19337827717718, −10.72859194268646, −10.42242529373046, −9.563880042381450, −9.469708282409816, −8.689798390687644, −8.285885291660858, −7.687954097968409, −7.568422948618657, −7.135439344096510, −6.238913245175758, −5.798184771421289, −4.925009290401063, −4.289172282532647, −4.014080062355272, −3.109207863547457, −2.485732821157619, −1.848419480442819, −0.8623981506274456, −0.4053576966910275, 0.4053576966910275, 0.8623981506274456, 1.848419480442819, 2.485732821157619, 3.109207863547457, 4.014080062355272, 4.289172282532647, 4.925009290401063, 5.798184771421289, 6.238913245175758, 7.135439344096510, 7.568422948618657, 7.687954097968409, 8.285885291660858, 8.689798390687644, 9.469708282409816, 9.563880042381450, 10.42242529373046, 10.72859194268646, 11.19337827717718, 11.67898875314258, 12.12217926716007, 12.73794348947713, 13.11834742812820, 13.87404577220165

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