Properties

Label 2-195195-1.1-c1-0-43
Degree $2$
Conductor $195195$
Sign $-1$
Analytic cond. $1558.63$
Root an. cond. $39.4796$
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 − 5-s + 6-s − 7-s − 3·8-s + 9-s − 10-s + 11-s − 12-s − 14-s − 15-s − 16-s + 2·17-s + 18-s − 4·19-s + 20-s − 21-s + 22-s − 3·24-s + 25-s + 27-s + 28-s + 6·29-s − 30-s + 5·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.218·21-s + 0.213·22-s − 0.612·24-s + 1/5·25-s + 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.182·30-s + 0.883·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195195\)    =    \(3 \cdot 5 \cdot 7 \cdot 11 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(1558.63\)
Root analytic conductor: \(39.4796\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{195195} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 195195,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 \)
good2 \( 1 - T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.30238257592488, −12.88641326386380, −12.49755861453971, −12.03341555785287, −11.73739967386692, −10.96521841303680, −10.43730385522223, −10.03837102313933, −9.440069092824382, −9.021131486676533, −8.599430628110787, −8.130846714578539, −7.705943434930058, −6.852526595714781, −6.679330933554179, −5.936959107819487, −5.526077518753959, −4.816544512948817, −4.318395992083873, −4.057677951191269, −3.225486391033027, −3.126413902704580, −2.360754353804629, −1.569028097077995, −0.7645601832375169, 0, 0.7645601832375169, 1.569028097077995, 2.360754353804629, 3.126413902704580, 3.225486391033027, 4.057677951191269, 4.318395992083873, 4.816544512948817, 5.526077518753959, 5.936959107819487, 6.679330933554179, 6.852526595714781, 7.705943434930058, 8.130846714578539, 8.599430628110787, 9.021131486676533, 9.440069092824382, 10.03837102313933, 10.43730385522223, 10.96521841303680, 11.73739967386692, 12.03341555785287, 12.49755861453971, 12.88641326386380, 13.30238257592488

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