Properties

Label 2-112710-1.1-c1-0-68
Degree $2$
Conductor $112710$
Sign $-1$
Analytic cond. $899.993$
Root an. cond. $29.9998$
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 − 2·7-s + 8-s + 9-s + 10-s − 3·11-s + 12-s − 13-s − 2·14-s + 15-s + 16-s + 18-s + 7·19-s + 20-s − 2·21-s − 3·22-s + 23-s + 24-s + 25-s − 26-s + 27-s − 2·28-s + 29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s + 0.288·12-s − 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s + 1.60·19-s + 0.223·20-s − 0.436·21-s − 0.639·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.377·28-s + 0.185·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112710\)    =    \(2 \cdot 3 \cdot 5 \cdot 13 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(899.993\)
Root analytic conductor: \(29.9998\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{112710} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 112710,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−13.81080097244476, −13.45100319802737, −12.96938499773985, −12.58058748147503, −12.12649141048841, −11.54094621709070, −10.91213132978815, −10.45690506190767, −9.908402286294865, −9.539167287196788, −9.095986750162742, −8.327029837151793, −7.914494592212952, −7.249155458586290, −6.882068569562667, −6.427516040950577, −5.511477162521078, −5.290261221183898, −4.897560451814415, −3.878447902841229, −3.467628064767964, −3.037103383812380, −2.393737114268289, −1.861025038909208, −1.024231009291884, 0, 1.024231009291884, 1.861025038909208, 2.393737114268289, 3.037103383812380, 3.467628064767964, 3.878447902841229, 4.897560451814415, 5.290261221183898, 5.511477162521078, 6.427516040950577, 6.882068569562667, 7.249155458586290, 7.914494592212952, 8.327029837151793, 9.095986750162742, 9.539167287196788, 9.908402286294865, 10.45690506190767, 10.91213132978815, 11.54094621709070, 12.12649141048841, 12.58058748147503, 12.96938499773985, 13.45100319802737, 13.81080097244476

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