Properties

Label 2-1-1.1-c15-0-0
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $1.42693$
Root an. cond. $1.19454$
Motivic weight $15$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 216·2-s − 3.34e3·3-s + 1.38e4·4-s + 5.21e4·5-s − 7.23e5·6-s + 2.82e6·7-s − 4.07e6·8-s − 3.13e6·9-s + 1.12e7·10-s + 2.05e7·11-s − 4.64e7·12-s − 1.90e8·13-s + 6.09e8·14-s − 1.74e8·15-s − 1.33e9·16-s + 1.64e9·17-s − 6.78e8·18-s + 1.56e9·19-s + 7.23e8·20-s − 9.44e9·21-s + 4.44e9·22-s + 9.45e9·23-s + 1.36e10·24-s − 2.78e10·25-s − 4.10e10·26-s + 5.85e10·27-s + 3.91e10·28-s + ⋯
L(s)  = 1  + 1.19·2-s − 0.883·3-s + 0.423·4-s + 0.298·5-s − 1.05·6-s + 1.29·7-s − 0.687·8-s − 0.218·9-s + 0.355·10-s + 0.318·11-s − 0.374·12-s − 0.840·13-s + 1.54·14-s − 0.263·15-s − 1.24·16-s + 0.973·17-s − 0.261·18-s + 0.401·19-s + 0.126·20-s − 1.14·21-s + 0.380·22-s + 0.578·23-s + 0.607·24-s − 0.911·25-s − 1.00·26-s + 1.07·27-s + 0.549·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(16-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+15/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(1.42693\)
Root analytic conductor: \(1.19454\)
Motivic weight: \(15\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :15/2),\ 1)\)

Particular Values

\(L(8)\) \(\approx\) \(1.520561669\)
\(L(\frac12)\) \(\approx\) \(1.520561669\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 27 p^{3} T + p^{15} T^{2} \)
3 \( 1 + 124 p^{3} T + p^{15} T^{2} \)
5 \( 1 - 10422 p T + p^{15} T^{2} \)
7 \( 1 - 403208 p T + p^{15} T^{2} \)
11 \( 1 - 1871532 p T + p^{15} T^{2} \)
13 \( 1 + 14621026 p T + p^{15} T^{2} \)
17 \( 1 - 1646527986 T + p^{15} T^{2} \)
19 \( 1 - 1563257180 T + p^{15} T^{2} \)
23 \( 1 - 9451116072 T + p^{15} T^{2} \)
29 \( 1 + 36902568330 T + p^{15} T^{2} \)
31 \( 1 - 71588483552 T + p^{15} T^{2} \)
37 \( 1 + 1033652081554 T + p^{15} T^{2} \)
41 \( 1 - 1641974018202 T + p^{15} T^{2} \)
43 \( 1 + 492403109308 T + p^{15} T^{2} \)
47 \( 1 + 3410684952624 T + p^{15} T^{2} \)
53 \( 1 - 6797151655902 T + p^{15} T^{2} \)
59 \( 1 - 167099268060 p T + p^{15} T^{2} \)
61 \( 1 - 4931842626902 T + p^{15} T^{2} \)
67 \( 1 + 28837826625364 T + p^{15} T^{2} \)
71 \( 1 - 125050114914552 T + p^{15} T^{2} \)
73 \( 1 + 82171455513478 T + p^{15} T^{2} \)
79 \( 1 + 25413078694480 T + p^{15} T^{2} \)
83 \( 1 + 281736730890468 T + p^{15} T^{2} \)
89 \( 1 - 715618564776810 T + p^{15} T^{2} \)
97 \( 1 - 612786136081826 T + p^{15} T^{2} \)
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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

−31.31083306980598975354369589119, −29.69484151629079972214366719026, −27.56748408646213265713341304974, −24.35638841200895818548143833510, −22.83359228719678482192637867477, −21.26112311281979572877543178918, −17.51673159287656967480964579195, −14.40110767072392844184488566452, −11.82395546014304725015828959152, −5.26502022758204893749954155127, 5.26502022758204893749954155127, 11.82395546014304725015828959152, 14.40110767072392844184488566452, 17.51673159287656967480964579195, 21.26112311281979572877543178918, 22.83359228719678482192637867477, 24.35638841200895818548143833510, 27.56748408646213265713341304974, 29.69484151629079972214366719026, 31.31083306980598975354369589119

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