Properties

Degree $2$
Conductor $8$
Sign $-1$
Motivic weight $7$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 84·3-s − 82·5-s − 456·7-s + 4.86e3·9-s − 2.52e3·11-s − 1.07e4·13-s + 6.88e3·15-s − 1.11e4·17-s + 4.12e3·19-s + 3.83e4·21-s + 8.17e4·23-s − 7.14e4·25-s − 2.25e5·27-s + 9.97e4·29-s − 4.04e4·31-s + 2.12e5·33-s + 3.73e4·35-s − 4.19e5·37-s + 9.05e5·39-s + 1.41e5·41-s − 6.90e5·43-s − 3.99e5·45-s − 6.82e5·47-s − 6.15e5·49-s + 9.36e5·51-s + 1.81e6·53-s + 2.06e5·55-s + ⋯
L(s)  = 1  − 1.79·3-s − 0.293·5-s − 0.502·7-s + 2.22·9-s − 0.571·11-s − 1.36·13-s + 0.526·15-s − 0.550·17-s + 0.137·19-s + 0.902·21-s + 1.40·23-s − 0.913·25-s − 2.20·27-s + 0.759·29-s − 0.244·31-s + 1.02·33-s + 0.147·35-s − 1.36·37-s + 2.44·39-s + 0.320·41-s − 1.32·43-s − 0.653·45-s − 0.958·47-s − 0.747·49-s + 0.988·51-s + 1.67·53-s + 0.167·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-1$
Motivic weight: \(7\)
Character: $\chi_{8} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 28 p T + p^{7} T^{2} \)
5 \( 1 + 82 T + p^{7} T^{2} \)
7 \( 1 + 456 T + p^{7} T^{2} \)
11 \( 1 + 2524 T + p^{7} T^{2} \)
13 \( 1 + 10778 T + p^{7} T^{2} \)
17 \( 1 + 11150 T + p^{7} T^{2} \)
19 \( 1 - 4124 T + p^{7} T^{2} \)
23 \( 1 - 81704 T + p^{7} T^{2} \)
29 \( 1 - 99798 T + p^{7} T^{2} \)
31 \( 1 + 40480 T + p^{7} T^{2} \)
37 \( 1 + 419442 T + p^{7} T^{2} \)
41 \( 1 - 141402 T + p^{7} T^{2} \)
43 \( 1 + 690428 T + p^{7} T^{2} \)
47 \( 1 + 682032 T + p^{7} T^{2} \)
53 \( 1 - 1813118 T + p^{7} T^{2} \)
59 \( 1 + 966028 T + p^{7} T^{2} \)
61 \( 1 - 1887670 T + p^{7} T^{2} \)
67 \( 1 - 2965868 T + p^{7} T^{2} \)
71 \( 1 + 2548232 T + p^{7} T^{2} \)
73 \( 1 + 1680326 T + p^{7} T^{2} \)
79 \( 1 - 4038064 T + p^{7} T^{2} \)
83 \( 1 + 5385764 T + p^{7} T^{2} \)
89 \( 1 + 6473046 T + p^{7} T^{2} \)
97 \( 1 + 6065758 T + p^{7} T^{2} \)
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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

−19.33539488014304505369483389471, −17.84118289171870269700157743928, −16.76571096552944371160403574876, −15.51849604957907210792176341191, −12.84884691759405480078086906011, −11.61154968464071773737550398844, −10.13804380825495415909992963500, −6.93776263614106550406017956614, −5.07707747027606474477459261931, 0, 5.07707747027606474477459261931, 6.93776263614106550406017956614, 10.13804380825495415909992963500, 11.61154968464071773737550398844, 12.84884691759405480078086906011, 15.51849604957907210792176341191, 16.76571096552944371160403574876, 17.84118289171870269700157743928, 19.33539488014304505369483389471

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