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Properties

Label 2-28e2-28.3-c1-0-5
Degree $2$
Conductor $784$
Sign $0.895 - 0.444i$
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 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.895 - 0.444i$
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: \((2,\ 784,\ (\ :1/2),\ 0.895 - 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43479 + 0.336117i\)
\(L(\frac12)\) \(\approx\) \(1.43479 + 0.336117i\)
\(L(\frac{3}{2})\) 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

−10.51081854527704736715709239042, −9.297607396541808730424968762583, −8.439237015467873143160342965635, −7.68241272580544848151433283162, −6.92251881430001579233400816378, −6.56873457262378479271924286932, −4.72113833715654742246823404797, −3.76917325712840530968217257658, −2.75280751449364011024900100461, −1.42204531492908734736656466423, 0.793224935602783770073573603726, 3.05901214614047173021603400116, 3.83498914311003899237019291344, 4.51326230884320034009920753224, 5.52663103598904008778325998195, 6.92927672538402624308263521748, 7.958483362041086165602091025108, 8.622022619969845210406207103158, 9.226783850815443263566038497664, 10.10998531859165168631929583354

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