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Properties

Label 2-28e2-112.53-c1-0-64
Degree $2$
Conductor $784$
Sign $-0.202 - 0.979i$
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  + (−1.21 − 0.723i)2-s + (−0.608 − 2.27i)3-s + (0.952 + 1.75i)4-s + (−0.400 + 1.49i)5-s + (−0.903 + 3.19i)6-s + (0.114 − 2.82i)8-s + (−2.18 + 1.26i)9-s + (1.56 − 1.52i)10-s + (−1.75 + 0.469i)11-s + (3.41 − 3.23i)12-s + (1.49 − 1.49i)13-s + 3.63·15-s + (−2.18 + 3.35i)16-s + (2.44 − 4.22i)17-s + (3.57 + 0.0482i)18-s + (−7.02 − 1.88i)19-s + ⋯
L(s)  = 1  + (−0.859 − 0.511i)2-s + (−0.351 − 1.31i)3-s + (0.476 + 0.879i)4-s + (−0.179 + 0.668i)5-s + (−0.368 + 1.30i)6-s + (0.0404 − 0.999i)8-s + (−0.729 + 0.421i)9-s + (0.495 − 0.482i)10-s + (−0.528 + 0.141i)11-s + (0.985 − 0.933i)12-s + (0.415 − 0.415i)13-s + 0.939·15-s + (−0.545 + 0.837i)16-s + (0.592 − 1.02i)17-s + (0.842 + 0.0113i)18-s + (−1.61 − 0.431i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.202 - 0.979i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.202 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000570802 + 0.000700754i\)
\(L(\frac12)\) \(\approx\) \(0.000570802 + 0.000700754i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.21 + 0.723i)T \)
7 \( 1 \)
good3 \( 1 + (0.608 + 2.27i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (0.400 - 1.49i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.75 - 0.469i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.49 + 1.49i)T - 13iT^{2} \)
17 \( 1 + (-2.44 + 4.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.02 + 1.88i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (6.87 - 3.97i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.18 - 4.18i)T - 29iT^{2} \)
31 \( 1 + (4.73 - 8.20i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.449 - 1.67i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 + (-2.27 - 2.27i)T + 43iT^{2} \)
47 \( 1 + (0.930 + 1.61i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.15 - 2.18i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.73 - 1.00i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.17 - 1.38i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.173 - 0.647i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 1.50iT - 71T^{2} \)
73 \( 1 + (-8.75 - 5.05i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.90 - 6.77i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.73 - 8.73i)T - 83iT^{2} \)
89 \( 1 + (-3.69 + 2.13i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.3T + 97T^{2} \)
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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

−9.775963242986757052298296505026, −8.702597056608081084954386740237, −7.85022558341895335856721082184, −7.18316107725227095474002586631, −6.63072627917379453347677253599, −5.47809885637608901586852753199, −3.70967250957028108582103951857, −2.59783222471970148865605141239, −1.54540916480064040354954516298, −0.00061122348892191412814809122, 1.98489904708861352437614281044, 3.93385915337731795721573875520, 4.62472042468565602963646486976, 5.77860067897353453049512916634, 6.29004871761989408757411853787, 7.919170516041948308735749026923, 8.337171968350242681087361205149, 9.308606841162162132217550350471, 9.969358462501303087214685214355, 10.69342539646447918105145351446

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