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Properties

Label 2-28e2-196.111-c1-0-20
Degree $2$
Conductor $784$
Sign $0.841 - 0.540i$
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  + (1.44 + 1.81i)3-s + (2.77 − 2.21i)5-s + (1.94 + 1.79i)7-s + (−0.528 + 2.31i)9-s + (4.57 − 1.04i)11-s + (−5.78 + 1.32i)13-s + (8.01 + 1.82i)15-s + (0.938 − 1.94i)17-s − 4.08·19-s + (−0.444 + 6.11i)21-s + (−1.07 − 2.23i)23-s + (1.68 − 7.38i)25-s + (1.30 − 0.628i)27-s + (3.94 + 1.90i)29-s − 6.72·31-s + ⋯
L(s)  = 1  + (0.834 + 1.04i)3-s + (1.24 − 0.988i)5-s + (0.734 + 0.678i)7-s + (−0.176 + 0.771i)9-s + (1.37 − 0.314i)11-s + (−1.60 + 0.366i)13-s + (2.06 + 0.472i)15-s + (0.227 − 0.472i)17-s − 0.936·19-s + (−0.0970 + 1.33i)21-s + (−0.224 − 0.467i)23-s + (0.337 − 1.47i)25-s + (0.251 − 0.121i)27-s + (0.732 + 0.352i)29-s − 1.20·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.841 - 0.540i$
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: \((2,\ 784,\ (\ :1/2),\ 0.841 - 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.53578 + 0.744015i\)
\(L(\frac12)\) \(\approx\) \(2.53578 + 0.744015i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.94 - 1.79i)T \)
good3 \( 1 + (-1.44 - 1.81i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (-2.77 + 2.21i)T + (1.11 - 4.87i)T^{2} \)
11 \( 1 + (-4.57 + 1.04i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (5.78 - 1.32i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (-0.938 + 1.94i)T + (-10.5 - 13.2i)T^{2} \)
19 \( 1 + 4.08T + 19T^{2} \)
23 \( 1 + (1.07 + 2.23i)T + (-14.3 + 17.9i)T^{2} \)
29 \( 1 + (-3.94 - 1.90i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + 6.72T + 31T^{2} \)
37 \( 1 + (8.04 + 3.87i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (-0.210 + 0.168i)T + (9.12 - 39.9i)T^{2} \)
43 \( 1 + (-3.94 - 3.14i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (0.0216 + 0.0950i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (10.4 - 5.01i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (1.91 - 2.40i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-1.79 + 3.73i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 - 4.19iT - 67T^{2} \)
71 \( 1 + (-3.95 - 8.20i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (1.51 + 0.345i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 - 14.9iT - 79T^{2} \)
83 \( 1 + (-0.959 + 4.20i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (16.4 + 3.74i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 - 15.2iT - 97T^{2} \)
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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

−9.980806047555240573488503700505, −9.334331458340468285788135111818, −8.991588112858088478348222837095, −8.284503982647340658749502190480, −6.84319581597591048943940303911, −5.66953033282279746074565621333, −4.85512137259719699696298289853, −4.17941183114277548149631396051, −2.65928715983560080006880923537, −1.68388368630776069618533641082, 1.64773173829150924731032993583, 2.18669753374069457102950988773, 3.43003361288274342992237099623, 4.82397692942509584352802876647, 6.14592835134308924405515199649, 6.95088032603682515130252268506, 7.41358073814052926692921013917, 8.399206000463806501618189326670, 9.415681432320927083108516247567, 10.14945261811430527841028468336

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