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Properties

Label 2-28e2-112.109-c1-0-7
Degree $2$
Conductor $784$
Sign $-0.615 - 0.788i$
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  + (−0.366 + 1.36i)2-s + (−2.73 + 0.732i)3-s + (−1.73 − i)4-s + (−2.73 − 0.732i)5-s − 4i·6-s + (2 − 1.99i)8-s + (4.33 − 2.5i)9-s + (2 − 3.46i)10-s + (−1.09 − 4.09i)11-s + (5.46 + 1.46i)12-s + 8·15-s + (1.99 + 3.46i)16-s + (−3 + 5.19i)17-s + (1.83 + 6.83i)18-s + (1.46 − 5.46i)19-s + (3.99 + 4i)20-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−1.57 + 0.422i)3-s + (−0.866 − 0.5i)4-s + (−1.22 − 0.327i)5-s − 1.63i·6-s + (0.707 − 0.707i)8-s + (1.44 − 0.833i)9-s + (0.632 − 1.09i)10-s + (−0.331 − 1.23i)11-s + (1.57 + 0.422i)12-s + 2.06·15-s + (0.499 + 0.866i)16-s + (−0.727 + 1.26i)17-s + (0.431 + 1.60i)18-s + (0.335 − 1.25i)19-s + (0.894 + 0.894i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.615 - 0.788i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.615 - 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.107754 + 0.220810i\)
\(L(\frac12)\) \(\approx\) \(0.107754 + 0.220810i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 - 1.36i)T \)
7 \( 1 \)
good3 \( 1 + (2.73 - 0.732i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (2.73 + 0.732i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.09 + 4.09i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.46 + 5.46i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.73 - i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1 + i)T + 29iT^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.09 + 1.09i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 2iT - 41T^{2} \)
43 \( 1 + (5 - 5i)T - 43iT^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.56 - 9.56i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.732 + 2.73i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.19 - 8.19i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-6.83 + 1.83i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 8iT - 71T^{2} \)
73 \( 1 + (-8.66 - 5i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7 - 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (-5.19 + 3i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−10.78897974402033536696377034103, −9.776499805602149807039209005735, −8.648223973964573130046510474770, −8.047727239075925460100087072753, −6.95299222741407454729971303404, −6.17378493065671124081982651026, −5.38815780835485413125841473125, −4.53840031563731185723751945666, −3.76687787982593349307910996315, −0.71599611012821947614742845731, 0.28576856576474805902487597112, 1.88419214746094262672510623099, 3.48264410196605567004355467369, 4.60806475060633600469588354625, 5.19428854032495937602208970643, 6.67518953996557328205203395397, 7.40023129399240621310203915093, 8.121068490125013537195873366887, 9.490357922310476849563238343787, 10.36038165689253034993555643800

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