Properties

Degree 2
Conductor $ 2^{4} \cdot 7^{2} $
Sign $0.266 - 0.963i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)3-s + (−0.499 + 0.866i)9-s + 4·13-s + (3 + 5.19i)17-s + (−1 + 1.73i)19-s + (2.5 + 4.33i)25-s + 4.00·27-s − 6·29-s + (2 + 3.46i)31-s + (−1 + 1.73i)37-s + (4 + 6.92i)39-s − 6·41-s − 8·43-s + (6 − 10.3i)47-s + (−6 + 10.3i)51-s + ⋯
L(s)  = 1  + (0.577 + 0.999i)3-s + (−0.166 + 0.288i)9-s + 1.10·13-s + (0.727 + 1.26i)17-s + (−0.229 + 0.397i)19-s + (0.5 + 0.866i)25-s + 0.769·27-s − 1.11·29-s + (0.359 + 0.622i)31-s + (−0.164 + 0.284i)37-s + (0.640 + 1.10i)39-s − 0.937·41-s − 1.21·43-s + (0.875 − 1.51i)47-s + (−0.840 + 1.45i)51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(784\)    =    \(2^{4} \cdot 7^{2}\)
\( \varepsilon \)  =  $0.266 - 0.963i$
motivic weight  =  \(1\)
character  :  $\chi_{784} (177, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 784,\ (\ :1/2),\ 0.266 - 0.963i)$
$L(1)$  $\approx$  $1.54848 + 1.17801i$
$L(\frac12)$  $\approx$  $1.54848 + 1.17801i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.34586967778408609110973286532, −9.707851007875754713392808853370, −8.666227011445378470037205584876, −8.329005074772606692744430179159, −7.00986246105369453839216729199, −5.97583003985655602163386035513, −4.99345678444641967411552541619, −3.74433925007353034501583347528, −3.40678987924838441751322896699, −1.62450956709714827554626904609, 1.04073954424139627738905894688, 2.33030116372514581166531426270, 3.35798242908381928438757915636, 4.68293780932496330597192328397, 5.86978903777549079274329894396, 6.81573567429567800497384834958, 7.54537155574140241934656446245, 8.332237409188002119946297277530, 9.064425253230624987589428692824, 10.06274249136189414685461143675

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