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Properties

Label 2-28e2-112.109-c1-0-38
Degree $2$
Conductor $784$
Sign $0.973 + 0.227i$
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  + (−1.27 − 0.613i)2-s + (2.68 − 0.719i)3-s + (1.24 + 1.56i)4-s + (0.857 + 0.229i)5-s + (−3.86 − 0.731i)6-s + (−0.627 − 2.75i)8-s + (4.09 − 2.36i)9-s + (−0.951 − 0.819i)10-s + (1.36 + 5.08i)11-s + (4.47 + 3.30i)12-s + (2.22 + 2.22i)13-s + 2.46·15-s + (−0.893 + 3.89i)16-s + (−1.62 + 2.81i)17-s + (−6.66 + 0.497i)18-s + (−0.671 + 2.50i)19-s + ⋯
L(s)  = 1  + (−0.900 − 0.434i)2-s + (1.54 − 0.415i)3-s + (0.623 + 0.782i)4-s + (0.383 + 0.102i)5-s + (−1.57 − 0.298i)6-s + (−0.221 − 0.975i)8-s + (1.36 − 0.787i)9-s + (−0.300 − 0.259i)10-s + (0.410 + 1.53i)11-s + (1.29 + 0.953i)12-s + (0.615 + 0.615i)13-s + 0.637·15-s + (−0.223 + 0.974i)16-s + (−0.394 + 0.683i)17-s + (−1.56 + 0.117i)18-s + (−0.154 + 0.575i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.227i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.973 + 0.227i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.973 + 0.227i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90690 - 0.219603i\)
\(L(\frac12)\) \(\approx\) \(1.90690 - 0.219603i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 + 0.613i)T \)
7 \( 1 \)
good3 \( 1 + (-2.68 + 0.719i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.857 - 0.229i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.36 - 5.08i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-2.22 - 2.22i)T + 13iT^{2} \)
17 \( 1 + (1.62 - 2.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.671 - 2.50i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-6.62 + 3.82i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.53 + 1.53i)T + 29iT^{2} \)
31 \( 1 + (-1.22 + 2.11i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.72 - 0.461i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.77iT - 41T^{2} \)
43 \( 1 + (7.51 - 7.51i)T - 43iT^{2} \)
47 \( 1 + (5.55 + 9.62i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.07 - 4.00i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.50 + 9.34i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.23 + 4.60i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.26 - 0.338i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 + (-5.55 - 3.20i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.64 - 6.30i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.96 + 1.96i)T + 83iT^{2} \)
89 \( 1 + (-0.576 + 0.332i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.34T + 97T^{2} \)
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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

−9.868972022696370155556707786022, −9.419226326844010862305619412400, −8.597757356752574368821597855000, −7.995870964545000065409975290340, −7.02724712556200041471088825011, −6.45266714686311978261812099171, −4.36018641876190086206718168260, −3.44788539460609411054464980081, −2.18876588917888162720925921407, −1.69538537957217306322297715414, 1.25650879306054415398590943897, 2.73012993033326055858622237657, 3.47140825640691579837381410151, 5.06391756880787027421002339431, 6.09797345308590254394005098529, 7.16417726563726268832702102458, 8.080261220241429182087916342917, 8.767752622710281828115437107115, 9.174100021445616704029594683503, 9.930226659242446637632951135439

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