Properties

Degree $2$
Conductor $5$
Sign $1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 2·3-s + 8·4-s − 5·5-s − 8·6-s + 6·7-s − 23·9-s + 20·10-s + 32·11-s + 16·12-s − 38·13-s − 24·14-s − 10·15-s − 64·16-s + 26·17-s + 92·18-s + 100·19-s − 40·20-s + 12·21-s − 128·22-s − 78·23-s + 25·25-s + 152·26-s − 100·27-s + 48·28-s − 50·29-s + 40·30-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.384·3-s + 4-s − 0.447·5-s − 0.544·6-s + 0.323·7-s − 0.851·9-s + 0.632·10-s + 0.877·11-s + 0.384·12-s − 0.810·13-s − 0.458·14-s − 0.172·15-s − 16-s + 0.370·17-s + 1.20·18-s + 1.20·19-s − 0.447·20-s + 0.124·21-s − 1.24·22-s − 0.707·23-s + 1/5·25-s + 1.14·26-s − 0.712·27-s + 0.323·28-s − 0.320·29-s + 0.243·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $1$
Motivic weight: \(3\)
Character: $\chi_{5} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.411861\)
\(L(\frac12)\) \(\approx\) \(0.411861\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + p T \)
good2 \( 1 + p^{2} T + p^{3} T^{2} \)
3 \( 1 - 2 T + p^{3} T^{2} \)
7 \( 1 - 6 T + p^{3} T^{2} \)
11 \( 1 - 32 T + p^{3} T^{2} \)
13 \( 1 + 38 T + p^{3} T^{2} \)
17 \( 1 - 26 T + p^{3} T^{2} \)
19 \( 1 - 100 T + p^{3} T^{2} \)
23 \( 1 + 78 T + p^{3} T^{2} \)
29 \( 1 + 50 T + p^{3} T^{2} \)
31 \( 1 + 108 T + p^{3} T^{2} \)
37 \( 1 - 266 T + p^{3} T^{2} \)
41 \( 1 - 22 T + p^{3} T^{2} \)
43 \( 1 - 442 T + p^{3} T^{2} \)
47 \( 1 + 514 T + p^{3} T^{2} \)
53 \( 1 - 2 T + p^{3} T^{2} \)
59 \( 1 - 500 T + p^{3} T^{2} \)
61 \( 1 + 518 T + p^{3} T^{2} \)
67 \( 1 - 126 T + p^{3} T^{2} \)
71 \( 1 - 412 T + p^{3} T^{2} \)
73 \( 1 + 878 T + p^{3} T^{2} \)
79 \( 1 - 600 T + p^{3} T^{2} \)
83 \( 1 - 282 T + p^{3} T^{2} \)
89 \( 1 + 150 T + p^{3} T^{2} \)
97 \( 1 - 386 T + p^{3} T^{2} \)
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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

−24.53719930434053859680587393842, −22.51370164112999450106246009488, −20.24253638460194980453802000495, −19.39406085946034311371495394382, −17.80936002011203375803360468238, −16.49202096766309066044503764833, −14.42700262096538081523197027989, −11.52156568135998034842009582479, −9.415016459828749011981823541780, −7.80368599340441310768273857061, 7.80368599340441310768273857061, 9.415016459828749011981823541780, 11.52156568135998034842009582479, 14.42700262096538081523197027989, 16.49202096766309066044503764833, 17.80936002011203375803360468238, 19.39406085946034311371495394382, 20.24253638460194980453802000495, 22.51370164112999450106246009488, 24.53719930434053859680587393842

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