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Properties

Label 2-28e2-49.15-c1-0-1
Degree $2$
Conductor $784$
Sign $-0.989 + 0.145i$
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  + (−2.08 + 2.61i)3-s + (1.04 − 1.30i)5-s + (1.76 − 1.97i)7-s + (−1.82 − 7.99i)9-s + (−0.694 + 3.04i)11-s + (−1.07 + 4.69i)13-s + (1.24 + 5.44i)15-s + (−2.86 + 1.37i)17-s − 8.26·19-s + (1.48 + 8.72i)21-s + (−0.850 − 0.409i)23-s + (0.492 + 2.15i)25-s + (15.6 + 7.55i)27-s + (−1.72 + 0.831i)29-s − 0.534·31-s + ⋯
L(s)  = 1  + (−1.20 + 1.51i)3-s + (0.465 − 0.583i)5-s + (0.665 − 0.746i)7-s + (−0.608 − 2.66i)9-s + (−0.209 + 0.918i)11-s + (−0.297 + 1.30i)13-s + (0.320 + 1.40i)15-s + (−0.694 + 0.334i)17-s − 1.89·19-s + (0.324 + 1.90i)21-s + (−0.177 − 0.0854i)23-s + (0.0985 + 0.431i)25-s + (3.01 + 1.45i)27-s + (−0.320 + 0.154i)29-s − 0.0960·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.989 + 0.145i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.989 + 0.145i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0300230 - 0.411872i\)
\(L(\frac12)\) \(\approx\) \(0.0300230 - 0.411872i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.76 + 1.97i)T \)
good3 \( 1 + (2.08 - 2.61i)T + (-0.667 - 2.92i)T^{2} \)
5 \( 1 + (-1.04 + 1.30i)T + (-1.11 - 4.87i)T^{2} \)
11 \( 1 + (0.694 - 3.04i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (1.07 - 4.69i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (2.86 - 1.37i)T + (10.5 - 13.2i)T^{2} \)
19 \( 1 + 8.26T + 19T^{2} \)
23 \( 1 + (0.850 + 0.409i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (1.72 - 0.831i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + 0.534T + 31T^{2} \)
37 \( 1 + (1.51 - 0.727i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + (1.26 - 1.58i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (5.13 + 6.43i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (2.65 - 11.6i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (-0.850 - 0.409i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (4.08 + 5.12i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (0.962 - 0.463i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 - 6.29T + 67T^{2} \)
71 \( 1 + (8.86 + 4.26i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-0.183 - 0.803i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 - 1.33T + 79T^{2} \)
83 \( 1 + (-1.98 - 8.71i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-2.21 - 9.70i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + 17.0T + 97T^{2} \)
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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

−10.78940304206955423049282049464, −9.968438084110257217611037234333, −9.311247320604315380390769227536, −8.525766848781246115030700760556, −6.96547072417861310834509482406, −6.24097190805224630325962010129, −5.05860387778196968986152856521, −4.53640302186722134752171691079, −3.97337740348677129501701711977, −1.79325978411976664922124175544, 0.22828187569762461306798700341, 1.89810900592515304484451491675, 2.69411201675673477634094831633, 4.84672322004277352923690753926, 5.72340906246746618326800881188, 6.22110330683043826960516612422, 7.05477647172386443798289395873, 8.103547529655821266018582012023, 8.567404267391349011321099057702, 10.32486182678546912117934925545

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