Properties

Label 2-97461-1.1-c1-0-12
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 2·5-s + 4·10-s − 11-s + 13-s − 4·16-s − 17-s − 2·19-s − 4·20-s + 2·22-s − 4·23-s − 25-s − 2·26-s − 5·31-s + 8·32-s + 2·34-s + 5·37-s + 4·38-s + 9·41-s − 13·43-s − 2·44-s + 8·46-s − 6·47-s + 2·50-s + 2·52-s + 3·53-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.894·5-s + 1.26·10-s − 0.301·11-s + 0.277·13-s − 16-s − 0.242·17-s − 0.458·19-s − 0.894·20-s + 0.426·22-s − 0.834·23-s − 1/5·25-s − 0.392·26-s − 0.898·31-s + 1.41·32-s + 0.342·34-s + 0.821·37-s + 0.648·38-s + 1.40·41-s − 1.98·43-s − 0.301·44-s + 1.17·46-s − 0.875·47-s + 0.282·50-s + 0.277·52-s + 0.412·53-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: \(1\)
Selberg data: \((2,\ 97461,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 - T \)
17 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−13.97281634652173, −13.56501845781600, −12.94184302457572, −12.50632933439204, −11.81172596657093, −11.44113168131928, −10.97042256068127, −10.57780959548652, −10.00073222504085, −9.525048915417879, −9.040945383599811, −8.505226478205663, −8.036629213470224, −7.671738154089454, −7.331267329152710, −6.429725543539203, −6.265959002500888, −5.299853667890346, −4.676194963875212, −4.109571556524308, −3.583094361628377, −2.796833436600531, −2.032672093995381, −1.530084828887845, −0.5778435724192416, 0, 0.5778435724192416, 1.530084828887845, 2.032672093995381, 2.796833436600531, 3.583094361628377, 4.109571556524308, 4.676194963875212, 5.299853667890346, 6.265959002500888, 6.429725543539203, 7.331267329152710, 7.671738154089454, 8.036629213470224, 8.505226478205663, 9.040945383599811, 9.525048915417879, 10.00073222504085, 10.57780959548652, 10.97042256068127, 11.44113168131928, 11.81172596657093, 12.50632933439204, 12.94184302457572, 13.56501845781600, 13.97281634652173

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