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Properties

Label 2-28e2-7.4-c1-0-6
Degree $2$
Conductor $784$
Sign $0.827 - 0.561i$
Analytic cond. $6.26027$
Root an. cond. $2.50205$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 1.22i)3-s + (1.41 + 2.44i)5-s + (0.500 + 0.866i)9-s + (−1 + 1.73i)11-s + 4·15-s + (−0.707 + 1.22i)17-s + (3.53 + 6.12i)19-s + (−2 − 3.46i)23-s + (−1.49 + 2.59i)25-s + 5.65·27-s + 2·29-s + (−4.24 + 7.34i)31-s + (1.41 + 2.44i)33-s + (−5 − 8.66i)37-s + 9.89·41-s + ⋯
L(s)  = 1  + (0.408 − 0.707i)3-s + (0.632 + 1.09i)5-s + (0.166 + 0.288i)9-s + (−0.301 + 0.522i)11-s + 1.03·15-s + (−0.171 + 0.297i)17-s + (0.811 + 1.40i)19-s + (−0.417 − 0.722i)23-s + (−0.299 + 0.519i)25-s + 1.08·27-s + 0.371·29-s + (−0.762 + 1.31i)31-s + (0.246 + 0.426i)33-s + (−0.821 − 1.42i)37-s + 1.54·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.827 - 0.561i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.827 - 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85763 + 0.570795i\)
\(L(\frac12)\) \(\approx\) \(1.85763 + 0.570795i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 1.22i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.41 - 2.44i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (0.707 - 1.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.53 - 6.12i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (4.24 - 7.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.89T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + (1.41 + 2.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.707 + 1.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.41 - 2.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (0.707 - 1.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 + (3.53 + 6.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.89T + 97T^{2} \)
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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

−10.41205677907272415785131474360, −9.687976577868852128537939728385, −8.519003655994952372041866873596, −7.64247431963880498591624727078, −7.00871055966131451992737695214, −6.18304243457929548922066132055, −5.14997551086115392925216488356, −3.70207059256534126130445520889, −2.53548501848702022901603781564, −1.73243044668630488788654761720, 1.00846479455911153282846250281, 2.63833122172139736969369907396, 3.82060634655989023985418983997, 4.85354118211895599616948036798, 5.51380781264771516297328365553, 6.67075843762315298985100363056, 7.84728745083316937669714874426, 8.810084327256655085895283736579, 9.387525358444879172699144142161, 9.854061268050495765854543145247

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