Properties

Label 2-28e2-112.109-c1-0-1
Degree $2$
Conductor $784$
Sign $0.376 - 0.926i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.766 − 1.18i)2-s + (−2.72 + 0.729i)3-s + (−0.826 + 1.82i)4-s + (−2.58 − 0.691i)5-s + (2.95 + 2.67i)6-s + (2.79 − 0.413i)8-s + (4.27 − 2.47i)9-s + (1.15 + 3.59i)10-s + (−1.43 − 5.36i)11-s + (0.920 − 5.56i)12-s + (−3.30 − 3.30i)13-s + 7.52·15-s + (−2.63 − 3.00i)16-s + (2.63 − 4.56i)17-s + (−6.21 − 3.19i)18-s + (−0.587 + 2.19i)19-s + ⋯
L(s)  = 1  + (−0.541 − 0.840i)2-s + (−1.57 + 0.421i)3-s + (−0.413 + 0.910i)4-s + (−1.15 − 0.309i)5-s + (1.20 + 1.09i)6-s + (0.989 − 0.146i)8-s + (1.42 − 0.823i)9-s + (0.365 + 1.13i)10-s + (−0.433 − 1.61i)11-s + (0.265 − 1.60i)12-s + (−0.915 − 0.915i)13-s + 1.94·15-s + (−0.658 − 0.752i)16-s + (0.639 − 1.10i)17-s + (−1.46 − 0.752i)18-s + (−0.134 + 0.502i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.376 - 0.926i$
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: \((2,\ 784,\ (\ :1/2),\ 0.376 - 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0267935 + 0.0180246i\)
\(L(\frac12)\) \(\approx\) \(0.0267935 + 0.0180246i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 1.18i)T \)
7 \( 1 \)
good3 \( 1 + (2.72 - 0.729i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (2.58 + 0.691i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.43 + 5.36i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (3.30 + 3.30i)T + 13iT^{2} \)
17 \( 1 + (-2.63 + 4.56i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.587 - 2.19i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.73 - 2.15i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.07 - 1.07i)T + 29iT^{2} \)
31 \( 1 + (2.24 - 3.88i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.534 - 0.143i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.35iT - 41T^{2} \)
43 \( 1 + (-1.67 + 1.67i)T - 43iT^{2} \)
47 \( 1 + (-0.245 - 0.425i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.11 + 7.90i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.946 - 3.53i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.95 - 7.30i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (12.4 - 3.33i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 16.2iT - 71T^{2} \)
73 \( 1 + (6.89 + 3.97i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.10 + 1.91i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.81 - 3.81i)T + 83iT^{2} \)
89 \( 1 + (-9.66 + 5.58i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.26T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.52529191868569316355791203340, −10.05524291741689640304429210427, −8.840993053586915533429515282810, −7.931814444981932941440404085797, −7.25130458403260116261667539948, −5.72517599946552167286308702744, −5.07157071542485315731812184361, −3.99572192132838401952349806543, −3.06162422957702287189077780792, −0.76212793809714843356854021458, 0.03811076563202241129484540193, 1.82908274749897874589179036031, 4.34485471058151378592881469489, 4.77415485267379752145948074693, 5.99431767691688605457294529303, 6.72210879290707404481459534795, 7.50617973166200429049843220301, 7.87810691011645326697881998573, 9.403486396044966577030543742333, 10.22379124377296769507666136007

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