Properties

Label 2-23805-1.1-c1-0-1
Degree $2$
Conductor $23805$
Sign $1$
Analytic cond. $190.083$
Root an. cond. $13.7870$
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-s − 4-s + 5-s − 3·8-s + 10-s − 4·11-s − 2·13-s − 16-s + 2·17-s − 4·19-s − 20-s − 4·22-s + 25-s − 2·26-s + 2·29-s + 5·32-s + 2·34-s + 10·37-s − 4·38-s − 3·40-s − 10·41-s − 4·43-s + 4·44-s − 8·47-s − 7·49-s + 50-s + 2·52-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.06·8-s + 0.316·10-s − 1.20·11-s − 0.554·13-s − 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.223·20-s − 0.852·22-s + 1/5·25-s − 0.392·26-s + 0.371·29-s + 0.883·32-s + 0.342·34-s + 1.64·37-s − 0.648·38-s − 0.474·40-s − 1.56·41-s − 0.609·43-s + 0.603·44-s − 1.16·47-s − 49-s + 0.141·50-s + 0.277·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23805\)    =    \(3^{2} \cdot 5 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(190.083\)
Root analytic conductor: \(13.7870\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 23805,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.348902336\)
\(L(\frac12)\) \(\approx\) \(1.348902336\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
23 \( 1 \)
good2 \( 1 - T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−15.14575427013386, −14.85436226984976, −14.38758666837038, −13.69770530662672, −13.27306503063558, −12.86610607744561, −12.41719837513393, −11.79332938203507, −11.11307990247955, −10.47075585033732, −9.818847103352574, −9.605683212980012, −8.747331681916898, −8.094760108755191, −7.831123059992707, −6.711275900956038, −6.329266209379614, −5.589605696603885, −4.945500990047709, −4.752556965139026, −3.817439445412623, −3.076682495998013, −2.559245032627228, −1.659699335672447, −0.4013638090388458, 0.4013638090388458, 1.659699335672447, 2.559245032627228, 3.076682495998013, 3.817439445412623, 4.752556965139026, 4.945500990047709, 5.589605696603885, 6.329266209379614, 6.711275900956038, 7.831123059992707, 8.094760108755191, 8.747331681916898, 9.605683212980012, 9.818847103352574, 10.47075585033732, 11.11307990247955, 11.79332938203507, 12.41719837513393, 12.86610607744561, 13.27306503063558, 13.69770530662672, 14.38758666837038, 14.85436226984976, 15.14575427013386

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