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Properties

Label 2-28e2-112.59-c1-0-65
Degree $2$
Conductor $784$
Sign $-0.419 + 0.907i$
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  + (1.11 − 0.867i)2-s + (0.379 + 1.41i)3-s + (0.495 − 1.93i)4-s + (−2.30 − 0.617i)5-s + (1.65 + 1.25i)6-s + (−1.12 − 2.59i)8-s + (0.737 − 0.425i)9-s + (−3.10 + 1.30i)10-s + (−0.732 − 2.73i)11-s + (2.93 − 0.0342i)12-s + (−4.80 − 4.80i)13-s − 3.49i·15-s + (−3.50 − 1.91i)16-s + (0.982 + 0.567i)17-s + (0.454 − 1.11i)18-s + (1.66 + 0.445i)19-s + ⋯
L(s)  = 1  + (0.789 − 0.613i)2-s + (0.219 + 0.817i)3-s + (0.247 − 0.968i)4-s + (−1.03 − 0.276i)5-s + (0.674 + 0.511i)6-s + (−0.398 − 0.917i)8-s + (0.245 − 0.141i)9-s + (−0.983 + 0.414i)10-s + (−0.220 − 0.823i)11-s + (0.846 − 0.00987i)12-s + (−1.33 − 1.33i)13-s − 0.902i·15-s + (−0.877 − 0.479i)16-s + (0.238 + 0.137i)17-s + (0.107 − 0.262i)18-s + (0.381 + 0.102i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.419 + 0.907i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.419 + 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.945642 - 1.47849i\)
\(L(\frac12)\) \(\approx\) \(0.945642 - 1.47849i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.11 + 0.867i)T \)
7 \( 1 \)
good3 \( 1 + (-0.379 - 1.41i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (2.30 + 0.617i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.732 + 2.73i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (4.80 + 4.80i)T + 13iT^{2} \)
17 \( 1 + (-0.982 - 0.567i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.66 - 0.445i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.668 - 1.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.26 + 5.26i)T - 29iT^{2} \)
31 \( 1 + (-4.15 + 7.20i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.53 - 5.72i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 1.63T + 41T^{2} \)
43 \( 1 + (1.33 - 1.33i)T - 43iT^{2} \)
47 \( 1 + (0.966 + 1.67i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.67 - 2.32i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (4.49 - 1.20i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.749 + 2.79i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (0.146 - 0.0392i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 + (3.12 - 5.40i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.90 - 2.25i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.71 + 9.71i)T - 83iT^{2} \)
89 \( 1 + (-5.80 - 10.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.23iT - 97T^{2} \)
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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

−10.03867688590031194976114962425, −9.619890117532405855687562741357, −8.281274754532757505608378142019, −7.57638286029176633276931835215, −6.23695063399661426793316344027, −5.13517726293556364567575476736, −4.47277270565977968769588002418, −3.52828634785935020420979666773, −2.78103767701351353384202431704, −0.65737558509632139103183811530, 2.02763058056618788962735549872, 3.15054136044983694418906299997, 4.45044000407085960335397994670, 4.94702773818260978387594246480, 6.60325665921736511677374436346, 7.13425122972846155903807521485, 7.57664520447759946507063036730, 8.453204942092757146379702659375, 9.583393083781811235779733317242, 10.78426899008766674337126644961

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