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Properties

Label 2-28e2-112.37-c1-0-20
Degree $2$
Conductor $784$
Sign $0.103 - 0.994i$
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  + (−1.02 + 0.972i)2-s + (−0.862 − 0.231i)3-s + (0.107 − 1.99i)4-s + (3.20 − 0.857i)5-s + (1.10 − 0.601i)6-s + (1.83 + 2.15i)8-s + (−1.90 − 1.10i)9-s + (−2.45 + 3.99i)10-s + (−0.799 + 2.98i)11-s + (−0.553 + 1.69i)12-s + (−4.03 + 4.03i)13-s − 2.95·15-s + (−3.97 − 0.428i)16-s + (−0.173 − 0.301i)17-s + (3.03 − 0.725i)18-s + (1.56 + 5.82i)19-s + ⋯
L(s)  = 1  + (−0.725 + 0.687i)2-s + (−0.497 − 0.133i)3-s + (0.0536 − 0.998i)4-s + (1.43 − 0.383i)5-s + (0.453 − 0.245i)6-s + (0.647 + 0.761i)8-s + (−0.636 − 0.367i)9-s + (−0.774 + 1.26i)10-s + (−0.240 + 0.899i)11-s + (−0.159 + 0.489i)12-s + (−1.11 + 1.11i)13-s − 0.763·15-s + (−0.994 − 0.107i)16-s + (−0.0421 − 0.0730i)17-s + (0.714 − 0.170i)18-s + (0.357 + 1.33i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.103 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.103 - 0.994i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.103 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.699311 + 0.630574i\)
\(L(\frac12)\) \(\approx\) \(0.699311 + 0.630574i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.02 - 0.972i)T \)
7 \( 1 \)
good3 \( 1 + (0.862 + 0.231i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-3.20 + 0.857i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.799 - 2.98i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (4.03 - 4.03i)T - 13iT^{2} \)
17 \( 1 + (0.173 + 0.301i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.56 - 5.82i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-5.40 - 3.11i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.21 + 1.21i)T - 29iT^{2} \)
31 \( 1 + (-0.631 - 1.09i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.77 + 2.35i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.68iT - 41T^{2} \)
43 \( 1 + (-4.05 - 4.05i)T + 43iT^{2} \)
47 \( 1 + (2.32 - 4.02i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.08 + 11.5i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.89 - 7.07i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.00195 + 0.00728i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (4.12 + 1.10i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 0.828iT - 71T^{2} \)
73 \( 1 + (-5.41 + 3.12i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.377 + 0.654i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.66 - 3.66i)T - 83iT^{2} \)
89 \( 1 + (-5.40 - 3.12i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.18T + 97T^{2} \)
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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

−10.07659136922498118568301891514, −9.556216364794747737035498775405, −9.081308118889802959483705081879, −7.85003237534778615242031342529, −6.92371562008995597692107180396, −6.15640460291154720706515761314, −5.41853101390120637515563302192, −4.67325557975623470092609188383, −2.42744203262995175466354886025, −1.35549309366076971189929313088, 0.67688597888468346463636886073, 2.55718793330940410309690506434, 2.87073150594380782536456059471, 4.84203848055307650609668081611, 5.62173257267594037748549464836, 6.61097687241150029369832837409, 7.63668936409689687862093545328, 8.670721963910900545234948670924, 9.401967884678706036356656123959, 10.26130191499010746480981866052

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