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Properties

Label 2-28e2-196.111-c1-0-14
Degree $2$
Conductor $784$
Sign $0.381 - 0.924i$
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  + (1.92 + 2.41i)3-s + (1.36 − 1.08i)5-s + (2.17 − 1.50i)7-s + (−1.46 + 6.40i)9-s + (−2.31 + 0.527i)11-s + (−1.77 + 0.404i)13-s + (5.25 + 1.19i)15-s + (−2.28 + 4.74i)17-s + 7.15·19-s + (7.84 + 2.35i)21-s + (−1.05 − 2.18i)23-s + (−0.438 + 1.92i)25-s + (−9.95 + 4.79i)27-s + (0.154 + 0.0742i)29-s + 6.24·31-s + ⋯
L(s)  = 1  + (1.11 + 1.39i)3-s + (0.608 − 0.485i)5-s + (0.821 − 0.569i)7-s + (−0.487 + 2.13i)9-s + (−0.697 + 0.159i)11-s + (−0.491 + 0.112i)13-s + (1.35 + 0.309i)15-s + (−0.554 + 1.15i)17-s + 1.64·19-s + (1.71 + 0.513i)21-s + (−0.219 − 0.455i)23-s + (−0.0876 + 0.384i)25-s + (−1.91 + 0.923i)27-s + (0.0286 + 0.0137i)29-s + 1.12·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.381 - 0.924i$
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: \((2,\ 784,\ (\ :1/2),\ 0.381 - 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10203 + 1.40706i\)
\(L(\frac12)\) \(\approx\) \(2.10203 + 1.40706i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 + 1.50i)T \)
good3 \( 1 + (-1.92 - 2.41i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (-1.36 + 1.08i)T + (1.11 - 4.87i)T^{2} \)
11 \( 1 + (2.31 - 0.527i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (1.77 - 0.404i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (2.28 - 4.74i)T + (-10.5 - 13.2i)T^{2} \)
19 \( 1 - 7.15T + 19T^{2} \)
23 \( 1 + (1.05 + 2.18i)T + (-14.3 + 17.9i)T^{2} \)
29 \( 1 + (-0.154 - 0.0742i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 - 6.24T + 31T^{2} \)
37 \( 1 + (-3.59 - 1.73i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (-8.90 + 7.09i)T + (9.12 - 39.9i)T^{2} \)
43 \( 1 + (6.95 + 5.54i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (1.92 + 8.41i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (8.28 - 3.98i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (3.40 - 4.27i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (0.0608 - 0.126i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 - 0.749iT - 67T^{2} \)
71 \( 1 + (-1.07 - 2.23i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (15.4 + 3.53i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + 6.44iT - 79T^{2} \)
83 \( 1 + (-2.99 + 13.1i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (6.19 + 1.41i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + 8.07iT - 97T^{2} \)
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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

−10.24605871358131496871083970697, −9.671578599568771755267145078062, −8.827639792843200484213571203942, −8.139560643975743109349880199913, −7.35613110333939538156858324084, −5.64400786731254361709049488499, −4.83184417081370030833032872668, −4.17653654106026646443405956114, −3.00006476357842157315400424309, −1.83103014080384761009807987864, 1.30243711236788386202853948662, 2.58762196130396978800079509678, 2.87009967412467212430089876331, 4.82841331645707374911798950424, 5.92779155575536524692500550596, 6.83162806112685553780568901819, 7.83470026273858914271299076393, 8.016133014732013988040866385641, 9.304926082652062304736509068412, 9.745575023832874453947329130767

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