↑

Properties

Label 2-28e2-112.93-c1-0-35
Degree $2$
Conductor $784$
Sign $0.908 - 0.417i$
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  + (0.0582 + 1.41i)2-s + (0.523 − 1.95i)3-s + (−1.99 + 0.164i)4-s + (0.256 + 0.959i)5-s + (2.78 + 0.625i)6-s + (−0.348 − 2.80i)8-s + (−0.941 − 0.543i)9-s + (−1.34 + 0.418i)10-s + (1.88 + 0.505i)11-s + (−0.721 + 3.97i)12-s + (2.10 + 2.10i)13-s + 2.00·15-s + (3.94 − 0.655i)16-s + (−2.83 − 4.91i)17-s + (0.713 − 1.36i)18-s + (−0.616 + 0.165i)19-s + ⋯
L(s)  = 1  + (0.0411 + 0.999i)2-s + (0.302 − 1.12i)3-s + (−0.996 + 0.0822i)4-s + (0.114 + 0.428i)5-s + (1.13 + 0.255i)6-s + (−0.123 − 0.992i)8-s + (−0.313 − 0.181i)9-s + (−0.423 + 0.132i)10-s + (0.569 + 0.152i)11-s + (−0.208 + 1.14i)12-s + (0.583 + 0.583i)13-s + 0.518·15-s + (0.986 − 0.163i)16-s + (−0.688 − 1.19i)17-s + (0.168 − 0.321i)18-s + (−0.141 + 0.0379i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.908 - 0.417i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (765, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.908 - 0.417i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67058 + 0.365363i\)
\(L(\frac12)\) \(\approx\) \(1.67058 + 0.365363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0582 - 1.41i)T \)
7 \( 1 \)
good3 \( 1 + (-0.523 + 1.95i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.256 - 0.959i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.88 - 0.505i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-2.10 - 2.10i)T + 13iT^{2} \)
17 \( 1 + (2.83 + 4.91i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.616 - 0.165i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-5.92 - 3.42i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.207 + 0.207i)T + 29iT^{2} \)
31 \( 1 + (-3.94 - 6.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.57 + 9.60i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.40iT - 41T^{2} \)
43 \( 1 + (-3.65 + 3.65i)T - 43iT^{2} \)
47 \( 1 + (-0.144 + 0.250i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.62 - 2.04i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-13.4 - 3.60i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.61 - 0.969i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (2.69 - 10.0i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 11.5iT - 71T^{2} \)
73 \( 1 + (0.310 - 0.179i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.84 - 6.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.424 - 0.424i)T + 83iT^{2} \)
89 \( 1 + (15.2 + 8.79i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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

−10.18057877106191162569029472524, −8.993852497456192694632626060298, −8.681280834288007785410980184674, −7.33532051428651365328929781729, −7.03331254680922043593597785726, −6.37161858183523274996796627355, −5.18730359081386851304734435448, −4.07017882724200711417928756339, −2.69381012580527467671205628151, −1.13342153363664495156081070983, 1.19139363242464241486658050296, 2.79695017901682449557151242347, 3.81360484238066291353925615990, 4.46823657260255976252595213215, 5.38286751317131989509294211309, 6.57784960136231606184367704743, 8.317582648381394211845042654049, 8.751302612671584187888437735638, 9.493521032042965352337271871786, 10.32268834372931420795497786346

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