Properties

Label 2-28e2-112.109-c1-0-17
Degree $2$
Conductor $784$
Sign $0.969 - 0.244i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.969 - 0.244i$
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: \((2,\ 784,\ (\ :1/2),\ 0.969 - 0.244i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.762015 + 0.0944462i\)
\(L(\frac12)\) \(\approx\) \(0.762015 + 0.0944462i\)
\(L(\frac{3}{2})\) 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

−10.58528820410941065197232891671, −9.105186474909132776102098995643, −8.636282111796482727205656618190, −8.089514151041692920815098465141, −7.34633041150352166762162421005, −6.07251256619522472893834904339, −4.43312098976730087855971417689, −3.67322204075888954530970047867, −2.74687119927540190665696233542, −1.13038861034554438045335962357, 0.56329745025311726048790979912, 2.67502446698994565659672376078, 3.76573925336767702054609089779, 4.91959446189868309311796510587, 6.08267631226901911805304017338, 7.12143696775799445275367588485, 7.74987596599791993779741404192, 8.471139178962390666514768473370, 9.123094865281363284318871077186, 10.13787281352283017552564583854

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