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Properties

Label 2-28e2-28.3-c1-0-0
Degree $2$
Conductor $784$
Sign $-0.832 - 0.553i$
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  + (−0.5 + 0.866i)3-s + (−1.5 + 0.866i)5-s + (1 + 1.73i)9-s + (1.5 + 0.866i)11-s − 1.73i·15-s + (4.5 + 2.59i)17-s + (−3.5 − 6.06i)19-s + (−7.5 + 4.33i)23-s + (−1 + 1.73i)25-s − 5·27-s − 6·29-s + (−2.5 + 4.33i)31-s + (−1.5 + 0.866i)33-s + (2.5 + 4.33i)37-s + 6.92i·41-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.670 + 0.387i)5-s + (0.333 + 0.577i)9-s + (0.452 + 0.261i)11-s − 0.447i·15-s + (1.09 + 0.630i)17-s + (−0.802 − 1.39i)19-s + (−1.56 + 0.902i)23-s + (−0.200 + 0.346i)25-s − 0.962·27-s − 1.11·29-s + (−0.449 + 0.777i)31-s + (−0.261 + 0.150i)33-s + (0.410 + 0.711i)37-s + 1.08i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.832 - 0.553i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.832 - 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.236263 + 0.781773i\)
\(L(\frac12)\) \(\approx\) \(0.236263 + 0.781773i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (-4.5 - 2.59i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.5 - 4.33i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.5 - 4.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.5 + 2.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (1.5 + 0.866i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.5 + 2.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-10.5 + 6.06i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−10.61134896921866550748012600908, −9.948182265675025666035636128605, −9.044780945958130956767712445732, −7.894984009004472794612556077037, −7.35402746501749637698505026235, −6.23184523140996318089149251445, −5.21034837874184783087873264824, −4.21309182963033013230675890583, −3.43706915351070291503683370059, −1.83573245524652134814498223316, 0.42420496715867480472003911201, 1.87760244114622577592496224355, 3.65090616517273560980214137767, 4.25488960492456462907084354138, 5.76340731267291607594274070710, 6.30631650614521880289639448094, 7.57132766433032385522981738611, 7.980563262373285162574847641449, 9.113931680129063772373857431901, 9.903381828491292382157415405779

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