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Properties

Label 2-28e2-112.109-c1-0-18
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  + (1.36 + 0.366i)2-s + (−1.36 + 0.366i)3-s + (1.73 + i)4-s + (−1.36 − 0.366i)5-s − 2·6-s + (1.99 + 2i)8-s + (−0.866 + 0.5i)9-s + (−1.73 − i)10-s + (0.366 + 1.36i)11-s + (−2.73 − 0.732i)12-s + (1 + i)13-s + 2·15-s + (1.99 + 3.46i)16-s + (−1 + 1.73i)17-s + (−1.36 + 0.366i)18-s + (−1.09 + 4.09i)19-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (−0.788 + 0.211i)3-s + (0.866 + 0.5i)4-s + (−0.610 − 0.163i)5-s − 0.816·6-s + (0.707 + 0.707i)8-s + (−0.288 + 0.166i)9-s + (−0.547 − 0.316i)10-s + (0.110 + 0.411i)11-s + (−0.788 − 0.211i)12-s + (0.277 + 0.277i)13-s + 0.516·15-s + (0.499 + 0.866i)16-s + (−0.242 + 0.420i)17-s + (−0.321 + 0.0862i)18-s + (−0.251 + 0.940i)19-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

\(L(1)\) \(\approx\) \(0.637806 + 1.30698i\)
\(L(\frac12)\) \(\approx\) \(0.637806 + 1.30698i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
7 \( 1 \)
good3 \( 1 + (1.36 - 0.366i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (1.36 + 0.366i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.366 - 1.36i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1 - i)T + 13iT^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.09 - 4.09i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (5.19 - 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 - 3i)T + 29iT^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.09 + 1.09i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-5 + 5i)T - 43iT^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.83 + 6.83i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.09 - 4.09i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.29 + 12.2i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-6.83 + 1.83i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 10iT - 71T^{2} \)
73 \( 1 + (-3.46 - 2i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1 - i)T + 83iT^{2} \)
89 \( 1 + (-3.46 + 2i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−10.86526175226527659985543761143, −10.09024142914215116793362636902, −8.613235494106915330104761898068, −7.901618339839388105951051010645, −6.89127760919081632955773886059, −6.01416840664914436388468621447, −5.28190650925256759172563825635, −4.30296248885310855361922526969, −3.54977982277054317555619917832, −1.94805754406740269319251515750, 0.56495862658599632052662303324, 2.42153094949907804958072285165, 3.59611010321157134259589652330, 4.51609685287121078803663308180, 5.60939877429752001293440665617, 6.24045477026544824353755734131, 7.09266228288039239077231175608, 8.069476125818346261938169996748, 9.255716778847940030909001990245, 10.44953683997457855633812806059

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