Properties

Degree $2$
Conductor $784$
Sign $-0.944 - 0.327i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 1.73i·5-s − 2·9-s + 1.73i·11-s + 1.73i·15-s − 5.19i·17-s − 7·19-s + 8.66i·23-s + 2.00·25-s + 5·27-s − 6·29-s − 5·31-s − 1.73i·33-s − 5·37-s + 6.92i·41-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.774i·5-s − 0.666·9-s + 0.522i·11-s + 0.447i·15-s − 1.26i·17-s − 1.60·19-s + 1.80i·23-s + 0.400·25-s + 0.962·27-s − 1.11·29-s − 0.898·31-s − 0.301i·33-s − 0.821·37-s + 1.08i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.944 - 0.327i$
Motivic weight: \(1\)
Character: $\chi_{784} (783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.944 - 0.327i)\)

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 \( 1 \)
7 \( 1 \)
good3 \( 1 + T + 3T^{2} \)
5 \( 1 + 1.73iT - 5T^{2} \)
11 \( 1 - 1.73iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 5.19iT - 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 - 8.66iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 - 3.46iT - 43T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + 9T + 59T^{2} \)
61 \( 1 + 8.66iT - 61T^{2} \)
67 \( 1 + 5.19iT - 67T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 - 1.73iT - 73T^{2} \)
79 \( 1 + 5.19iT - 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 - 12.1iT - 89T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−9.643056866037542507659769792663, −9.112175241764027222849099826751, −8.147538782730181230325227529983, −7.20485023568168009974022485961, −6.18300725804138709988492524008, −5.25680477505961232193244877553, −4.62396189926426788124148700380, −3.24897452515178296615306938291, −1.73270708803653011484508016380, 0, 2.12425479085155313175740843581, 3.32709934635960876556728387188, 4.43140932129377275546216121694, 5.72246113567273475562701106067, 6.29576725602317367963590051517, 7.10152964215978916313235657320, 8.424139223477214800317221720020, 8.811387563063235717931825421809, 10.35604132320528957203170963425

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