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Properties

Label 2-28e2-196.111-c1-0-16
Degree $2$
Conductor $784$
Sign $0.951 - 0.307i$
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  + (0.881 + 1.10i)3-s + (2.63 − 2.09i)5-s + (−0.910 + 2.48i)7-s + (0.222 − 0.976i)9-s + (−1.56 + 0.357i)11-s + (5.11 − 1.16i)13-s + (4.64 + 1.05i)15-s + (−2.40 + 4.99i)17-s + 4.03·19-s + (−3.54 + 1.18i)21-s + (−1.36 − 2.83i)23-s + (1.41 − 6.18i)25-s + (5.09 − 2.45i)27-s + (5.93 + 2.85i)29-s − 0.690·31-s + ⋯
L(s)  = 1  + (0.508 + 0.638i)3-s + (1.17 − 0.939i)5-s + (−0.344 + 0.938i)7-s + (0.0743 − 0.325i)9-s + (−0.472 + 0.107i)11-s + (1.41 − 0.323i)13-s + (1.19 + 0.273i)15-s + (−0.583 + 1.21i)17-s + 0.925·19-s + (−0.774 + 0.258i)21-s + (−0.284 − 0.590i)23-s + (0.282 − 1.23i)25-s + (0.980 − 0.472i)27-s + (1.10 + 0.531i)29-s − 0.124·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.951 - 0.307i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.951 - 0.307i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21251 + 0.348750i\)
\(L(\frac12)\) \(\approx\) \(2.21251 + 0.348750i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.910 - 2.48i)T \)
good3 \( 1 + (-0.881 - 1.10i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (-2.63 + 2.09i)T + (1.11 - 4.87i)T^{2} \)
11 \( 1 + (1.56 - 0.357i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (-5.11 + 1.16i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (2.40 - 4.99i)T + (-10.5 - 13.2i)T^{2} \)
19 \( 1 - 4.03T + 19T^{2} \)
23 \( 1 + (1.36 + 2.83i)T + (-14.3 + 17.9i)T^{2} \)
29 \( 1 + (-5.93 - 2.85i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + 0.690T + 31T^{2} \)
37 \( 1 + (4.31 + 2.07i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (-2.04 + 1.63i)T + (9.12 - 39.9i)T^{2} \)
43 \( 1 + (5.73 + 4.57i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (-1.83 - 8.03i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (4.20 - 2.02i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (-5.62 + 7.05i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (4.60 - 9.56i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + 9.32iT - 67T^{2} \)
71 \( 1 + (-1.34 - 2.79i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (3.73 + 0.852i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 - 7.64iT - 79T^{2} \)
83 \( 1 + (3.67 - 16.1i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-4.52 - 1.03i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + 18.7iT - 97T^{2} \)
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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

−10.11774094876949765192737844426, −9.399697731704046965112197632223, −8.701016362419175549807088876989, −8.344423142204742760468687584070, −6.52496107233608239650889513193, −5.85998379346936275241672937327, −5.04870080948568854521702135540, −3.86191095691302066616856992615, −2.74861939410073844751275384704, −1.44021210325141122352425720308, 1.38452253989426301228841385360, 2.56953594723981072930836555410, 3.43211348893771068932182679394, 4.93742034112212410041428944498, 6.13140830958566194310247092580, 6.83231245371382890067710420745, 7.48916105262788174834798661656, 8.487191159469989386565097666879, 9.544623343983368906410553833469, 10.23828649014696052861691597496

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