Properties

Label 2-98315-1.1-c1-0-1
Degree $2$
Conductor $98315$
Sign $1$
Analytic cond. $785.049$
Root an. cond. $28.0187$
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  − 3-s − 2·4-s − 5-s − 7-s − 2·9-s − 4·11-s + 2·12-s + 3·13-s + 15-s + 4·16-s + 2·17-s + 8·19-s + 2·20-s + 21-s + 4·23-s + 25-s + 5·27-s + 2·28-s − 6·29-s + 4·33-s + 35-s + 4·36-s + 8·37-s − 3·39-s − 6·41-s − 6·43-s + 8·44-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.447·5-s − 0.377·7-s − 2/3·9-s − 1.20·11-s + 0.577·12-s + 0.832·13-s + 0.258·15-s + 16-s + 0.485·17-s + 1.83·19-s + 0.447·20-s + 0.218·21-s + 0.834·23-s + 1/5·25-s + 0.962·27-s + 0.377·28-s − 1.11·29-s + 0.696·33-s + 0.169·35-s + 2/3·36-s + 1.31·37-s − 0.480·39-s − 0.937·41-s − 0.914·43-s + 1.20·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98315\)    =    \(5 \cdot 7 \cdot 53^{2}\)
Sign: $1$
Analytic conductor: \(785.049\)
Root analytic conductor: \(28.0187\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 98315,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.219311292\)
\(L(\frac12)\) \(\approx\) \(1.219311292\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
7 \( 1 + T \)
53 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 + T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.54299583634362, −13.32916906603756, −12.94063961282603, −12.23674583140926, −11.88414762227406, −11.21794340642907, −10.98881467946236, −10.27666526258266, −9.857496214690317, −9.229831851765473, −8.944626649966969, −8.094771590464993, −7.894597504667648, −7.423153617204636, −6.508104141102858, −6.082588496181260, −5.399882768341117, −5.042314745404408, −4.793304346976916, −3.625870432012474, −3.391401406935021, −2.958559127958289, −1.872962470799837, −0.7982989736560958, −0.5409082316770295, 0.5409082316770295, 0.7982989736560958, 1.872962470799837, 2.958559127958289, 3.391401406935021, 3.625870432012474, 4.793304346976916, 5.042314745404408, 5.399882768341117, 6.082588496181260, 6.508104141102858, 7.423153617204636, 7.894597504667648, 8.094771590464993, 8.944626649966969, 9.229831851765473, 9.857496214690317, 10.27666526258266, 10.98881467946236, 11.21794340642907, 11.88414762227406, 12.23674583140926, 12.94063961282603, 13.32916906603756, 13.54299583634362

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