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Properties

Label 2-28e2-112.53-c1-0-26
Degree $2$
Conductor $784$
Sign $-0.999 + 0.0349i$
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  + (1.07 + 0.922i)2-s + (0.781 + 2.91i)3-s + (0.299 + 1.97i)4-s + (−0.199 + 0.745i)5-s + (−1.85 + 3.84i)6-s + (−1.50 + 2.39i)8-s + (−5.28 + 3.05i)9-s + (−0.901 + 0.615i)10-s + (1.24 − 0.333i)11-s + (−5.52 + 2.41i)12-s + (0.919 − 0.919i)13-s − 2.32·15-s + (−3.82 + 1.18i)16-s + (3.95 − 6.85i)17-s + (−8.48 − 1.60i)18-s + (1.78 + 0.478i)19-s + ⋯
L(s)  = 1  + (0.758 + 0.651i)2-s + (0.450 + 1.68i)3-s + (0.149 + 0.988i)4-s + (−0.0893 + 0.333i)5-s + (−0.755 + 1.57i)6-s + (−0.530 + 0.847i)8-s + (−1.76 + 1.01i)9-s + (−0.285 + 0.194i)10-s + (0.374 − 0.100i)11-s + (−1.59 + 0.698i)12-s + (0.254 − 0.254i)13-s − 0.601·15-s + (−0.955 + 0.296i)16-s + (0.959 − 1.66i)17-s + (−2.00 − 0.377i)18-s + (0.409 + 0.109i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.999 + 0.0349i$
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: \((2,\ 784,\ (\ :1/2),\ -0.999 + 0.0349i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0450800 - 2.57571i\)
\(L(\frac12)\) \(\approx\) \(0.0450800 - 2.57571i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.07 - 0.922i)T \)
7 \( 1 \)
good3 \( 1 + (-0.781 - 2.91i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (0.199 - 0.745i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.24 + 0.333i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.919 + 0.919i)T - 13iT^{2} \)
17 \( 1 + (-3.95 + 6.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.78 - 0.478i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.33 - 1.92i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.25 + 5.25i)T - 29iT^{2} \)
31 \( 1 + (2.44 - 4.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.343 + 1.28i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.84iT - 41T^{2} \)
43 \( 1 + (-0.585 - 0.585i)T + 43iT^{2} \)
47 \( 1 + (-2.86 - 4.95i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.51 - 2.54i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-8.93 + 2.39i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.38 + 1.71i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.59 + 5.94i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 1.99iT - 71T^{2} \)
73 \( 1 + (6.69 + 3.86i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.63 + 8.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.78 + 4.78i)T - 83iT^{2} \)
89 \( 1 + (1.84 - 1.06i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.01T + 97T^{2} \)
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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

−10.70333968996451564624240185166, −9.716402718967845156042729376128, −9.122186585458658599226870741229, −8.140625873429346727507694741389, −7.32606676611880259404419071354, −6.06940567347659614356608985790, −5.16419003217298452936229196247, −4.47224208022357977673514954629, −3.39670454128117286819492897732, −2.90964053107212507572421145996, 1.04841169713507074104795098046, 1.89805254170668463503270041553, 3.07651533020985074574749628223, 4.13753546784964906334085644777, 5.56866375101768214307055098443, 6.33741676443512812977586353712, 7.07956880449958751517333718195, 8.185060999850002579365253197562, 8.814941796962327143212368688363, 9.999193734841132635734167837301

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