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Properties

Label 2-28e2-112.109-c1-0-68
Degree $2$
Conductor $784$
Sign $0.475 + 0.879i$
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  + (1.25 − 0.655i)2-s + (2.01 − 0.538i)3-s + (1.14 − 1.64i)4-s + (0.483 + 0.129i)5-s + (2.16 − 1.99i)6-s + (0.353 − 2.80i)8-s + (1.15 − 0.667i)9-s + (0.690 − 0.154i)10-s + (0.122 + 0.456i)11-s + (1.40 − 3.91i)12-s + (3.17 + 3.17i)13-s + 1.04·15-s + (−1.39 − 3.74i)16-s + (0.646 − 1.11i)17-s + (1.01 − 1.59i)18-s + (−0.963 + 3.59i)19-s + ⋯
L(s)  = 1  + (0.886 − 0.463i)2-s + (1.16 − 0.311i)3-s + (0.570 − 0.821i)4-s + (0.216 + 0.0579i)5-s + (0.884 − 0.813i)6-s + (0.125 − 0.992i)8-s + (0.385 − 0.222i)9-s + (0.218 − 0.0488i)10-s + (0.0368 + 0.137i)11-s + (0.406 − 1.13i)12-s + (0.881 + 0.881i)13-s + 0.269·15-s + (−0.348 − 0.937i)16-s + (0.156 − 0.271i)17-s + (0.238 − 0.375i)18-s + (−0.221 + 0.824i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.475 + 0.879i$
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: \((2,\ 784,\ (\ :1/2),\ 0.475 + 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.26451 - 1.94632i\)
\(L(\frac12)\) \(\approx\) \(3.26451 - 1.94632i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.25 + 0.655i)T \)
7 \( 1 \)
good3 \( 1 + (-2.01 + 0.538i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.483 - 0.129i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.122 - 0.456i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-3.17 - 3.17i)T + 13iT^{2} \)
17 \( 1 + (-0.646 + 1.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.963 - 3.59i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.26 - 1.30i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.98 + 6.98i)T + 29iT^{2} \)
31 \( 1 + (-4.17 + 7.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.53 - 1.21i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 9.93iT - 41T^{2} \)
43 \( 1 + (7.61 - 7.61i)T - 43iT^{2} \)
47 \( 1 + (2.29 + 3.98i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.44 + 5.38i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.34 - 8.73i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.59 - 5.94i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-13.6 + 3.65i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.62iT - 71T^{2} \)
73 \( 1 + (-0.482 - 0.278i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.744 - 1.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.47 + 1.47i)T + 83iT^{2} \)
89 \( 1 + (-10.6 + 6.14i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.50T + 97T^{2} \)
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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

−9.851060364169886170597499588490, −9.648864065639307954111096676115, −8.335907448998329612410787747070, −7.66894914984987355990296854356, −6.44023676439583953227930739491, −5.79668268976651026916224596401, −4.33131837802675042801260330474, −3.63005604303695289242005330043, −2.46860830927600097575610697654, −1.65354666267759970674101552297, 2.05568608714693906671949111368, 3.27126962439908729571081028493, 3.75904896151306922416335628265, 5.07041132161586381191848668883, 5.92741526369580842484322830853, 6.95862299332827757437983887855, 7.977894266991893457895571448856, 8.556652651417067748669770660449, 9.321657437338496978522517435388, 10.51794814034655832272091281807

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