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Properties

Label 2-28e2-112.3-c1-0-41
Degree $2$
Conductor $784$
Sign $-0.809 + 0.587i$
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  + (0.613 − 1.27i)2-s + (−2.05 + 0.551i)3-s + (−1.24 − 1.56i)4-s + (−0.929 + 3.47i)5-s + (−0.560 + 2.95i)6-s + (−2.75 + 0.627i)8-s + (1.33 − 0.768i)9-s + (3.85 + 3.31i)10-s + (2.73 − 0.732i)11-s + (3.42 + 2.53i)12-s + (1.17 − 1.17i)13-s − 7.65i·15-s + (−0.893 + 3.89i)16-s + (−5.31 − 3.06i)17-s + (−0.162 − 2.16i)18-s + (0.358 − 1.33i)19-s + ⋯
L(s)  = 1  + (0.434 − 0.900i)2-s + (−1.18 + 0.318i)3-s + (−0.623 − 0.782i)4-s + (−0.415 + 1.55i)5-s + (−0.228 + 1.20i)6-s + (−0.975 + 0.221i)8-s + (0.443 − 0.256i)9-s + (1.21 + 1.04i)10-s + (0.823 − 0.220i)11-s + (0.989 + 0.730i)12-s + (0.326 − 0.326i)13-s − 1.97i·15-s + (−0.223 + 0.974i)16-s + (−1.28 − 0.743i)17-s + (−0.0381 − 0.510i)18-s + (0.0822 − 0.307i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.809 + 0.587i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.809 + 0.587i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.149738 - 0.461288i\)
\(L(\frac12)\) \(\approx\) \(0.149738 - 0.461288i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.613 + 1.27i)T \)
7 \( 1 \)
good3 \( 1 + (2.05 - 0.551i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (0.929 - 3.47i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.73 + 0.732i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.17 + 1.17i)T - 13iT^{2} \)
17 \( 1 + (5.31 + 3.06i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.358 + 1.33i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.103 + 0.179i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.46 + 3.46i)T + 29iT^{2} \)
31 \( 1 + (-2.87 + 4.97i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.349 - 0.0935i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 6.75T + 41T^{2} \)
43 \( 1 + (-0.207 - 0.207i)T + 43iT^{2} \)
47 \( 1 + (5.46 + 9.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.63 + 9.82i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.743 - 2.77i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-10.2 - 2.74i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.358 - 1.33i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 2.09T + 71T^{2} \)
73 \( 1 + (1.36 - 2.36i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-9.55 + 5.51i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.27 - 2.27i)T + 83iT^{2} \)
89 \( 1 + (7.04 + 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.83iT - 97T^{2} \)
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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

−10.32012434643160453894887634779, −9.586943910355751844861824613901, −8.384917943654674810532991135884, −6.86402686834424658585802790793, −6.39031002799102997279262854051, −5.41592016298478654191717506086, −4.34127559119805449688003969850, −3.45986460022099895217655726551, −2.33575541991433020853041013525, −0.26917374535714871020482173686, 1.30405926218190254224411402031, 3.80111077750509926929789710456, 4.63295614697808611496327116044, 5.28902156714544215720124638237, 6.27689009302564681197906149642, 6.83062353849713673935094779216, 8.061117594097145369612148070549, 8.774176266924787547647124273966, 9.404471435744667820408012850955, 10.93922068195232317898210459673

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