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Properties

Label 2-28e2-28.3-c1-0-16
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.32 − 2.29i)3-s + (1.5 − 0.866i)5-s + (−2 − 3.46i)9-s + (3.96 + 2.29i)11-s − 3.46i·13-s − 4.58i·15-s + (−4.5 − 2.59i)17-s + (1.32 + 2.29i)19-s + (3.96 − 2.29i)23-s + (−1 + 1.73i)25-s − 2.64·27-s + (−1.32 + 2.29i)31-s + (10.5 − 6.06i)33-s + (−3.5 − 6.06i)37-s + (−7.93 − 4.58i)39-s + ⋯
L(s)  = 1  + (0.763 − 1.32i)3-s + (0.670 − 0.387i)5-s + (−0.666 − 1.15i)9-s + (1.19 + 0.690i)11-s − 0.960i·13-s − 1.18i·15-s + (−1.09 − 0.630i)17-s + (0.303 + 0.525i)19-s + (0.827 − 0.477i)23-s + (−0.200 + 0.346i)25-s − 0.509·27-s + (−0.237 + 0.411i)31-s + (1.82 − 1.05i)33-s + (−0.575 − 0.996i)37-s + (−1.27 − 0.733i)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: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.0633 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55311 - 1.65479i\)
\(L(\frac12)\) \(\approx\) \(1.55311 - 1.65479i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 + 2.29i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.96 - 2.29i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (4.5 + 2.59i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.32 - 2.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.96 + 2.29i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (1.32 - 2.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 - 9.16iT - 43T^{2} \)
47 \( 1 + (3.96 + 6.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.96 - 6.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.5 - 0.866i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.96 + 2.29i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.16iT - 71T^{2} \)
73 \( 1 + (-4.5 - 2.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.96 + 2.29i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.46iT - 97T^{2} \)
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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

−9.772800670900643337370473544842, −9.091847787259854855872006957377, −8.407546065498826505984483834477, −7.37253426368416677063296260948, −6.80732975759527254443882818079, −5.86239574785644351083548208802, −4.67177857677793635369546240818, −3.22709230755673541171014403680, −2.10044089283890058358601052338, −1.17664737245856564046973385862, 1.93468072503864251053803432649, 3.19971689517222703145765195368, 4.02137749741887733056354672407, 4.89766465455579452480983996719, 6.18056397595386981578727254036, 6.88676255904892415173507813346, 8.353837576056802013900867765567, 9.173998437115036129109699556750, 9.374691113669426112223084786880, 10.43219698743277457019295800849

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