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Properties

Label 2-28e2-28.3-c1-0-9
Degree $2$
Conductor $784$
Sign $-0.0633 + 0.997i$
Analytic cond. $6.26027$
Root an. cond. $2.50205$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)3-s + (−3 + 1.73i)5-s + (−0.499 − 0.866i)9-s + (−3 − 1.73i)11-s + 3.46i·13-s − 6.92i·15-s + (−1 − 1.73i)19-s + (−3 + 1.73i)23-s + (3.5 − 6.06i)25-s − 4.00·27-s + 6·29-s + (4 − 6.92i)31-s + (6 − 3.46i)33-s + (1 + 1.73i)37-s + (−5.99 − 3.46i)39-s + ⋯
L(s)  = 1  + (−0.577 + 0.999i)3-s + (−1.34 + 0.774i)5-s + (−0.166 − 0.288i)9-s + (−0.904 − 0.522i)11-s + 0.960i·13-s − 1.78i·15-s + (−0.229 − 0.397i)19-s + (−0.625 + 0.361i)23-s + (0.700 − 1.21i)25-s − 0.769·27-s + 1.11·29-s + (0.718 − 1.24i)31-s + (1.04 − 0.603i)33-s + (0.164 + 0.284i)37-s + (−0.960 − 0.554i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.0633 + 0.997i$
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: \(1\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.0633 + 0.997i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (3 - 1.73i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 - 1.73i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 + 10.3iT - 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 + 1.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3 + 1.73i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (6 + 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3 + 1.73i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (6 - 3.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−10.45064092404618624576041391837, −9.363460232001938192223691988724, −8.277330397860609314401042911877, −7.53938533565767705793031433471, −6.58641186022314235151630850492, −5.49229157437087386847296677144, −4.41482417676933497181119838178, −3.85077914416148281400819986736, −2.62881084871766801544384656958, 0, 1.22966827996759562250840495380, 2.94662097412163382535566968564, 4.30963641674011181378805949560, 5.12899649451736412572202856689, 6.23313657320194833803067148870, 7.16918649032956188654591363765, 8.135106824139695214037094799330, 8.197497193922523645982764813321, 9.755912146266170157356787980229

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