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Properties

Label 2-28e2-112.53-c1-0-63
Degree $2$
Conductor $784$
Sign $0.411 + 0.911i$
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  + (0.867 + 1.11i)2-s + (−0.442 − 1.65i)3-s + (−0.495 + 1.93i)4-s + (0.949 − 3.54i)5-s + (1.45 − 1.92i)6-s + (−2.59 + 1.12i)8-s + (0.0695 − 0.0401i)9-s + (4.78 − 2.01i)10-s + (1.47 − 0.395i)11-s + (3.41 − 0.0398i)12-s + (−2.97 + 2.97i)13-s − 6.26·15-s + (−3.50 − 1.91i)16-s + (3.59 − 6.22i)17-s + (0.105 + 0.0428i)18-s + (−3.08 − 0.826i)19-s + ⋯
L(s)  = 1  + (0.613 + 0.789i)2-s + (−0.255 − 0.952i)3-s + (−0.247 + 0.968i)4-s + (0.424 − 1.58i)5-s + (0.595 − 0.786i)6-s + (−0.917 + 0.398i)8-s + (0.0231 − 0.0133i)9-s + (1.51 − 0.636i)10-s + (0.444 − 0.119i)11-s + (0.986 − 0.0115i)12-s + (−0.826 + 0.826i)13-s − 1.61·15-s + (−0.877 − 0.479i)16-s + (0.871 − 1.50i)17-s + (0.0247 + 0.0100i)18-s + (−0.707 − 0.189i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.411 + 0.911i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.411 + 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55752 - 1.00543i\)
\(L(\frac12)\) \(\approx\) \(1.55752 - 1.00543i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.867 - 1.11i)T \)
7 \( 1 \)
good3 \( 1 + (0.442 + 1.65i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (-0.949 + 3.54i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.47 + 0.395i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (2.97 - 2.97i)T - 13iT^{2} \)
17 \( 1 + (-3.59 + 6.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.08 + 0.826i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.02 + 1.74i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.851 + 0.851i)T - 29iT^{2} \)
31 \( 1 + (1.97 - 3.42i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.17 + 8.10i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.67iT - 41T^{2} \)
43 \( 1 + (4.25 + 4.25i)T + 43iT^{2} \)
47 \( 1 + (-1.17 - 2.03i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.66 + 1.25i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.27 + 1.41i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.92 - 0.514i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.08 - 7.79i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 14.4iT - 71T^{2} \)
73 \( 1 + (-2.88 - 1.66i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.90 - 13.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.20 - 1.20i)T - 83iT^{2} \)
89 \( 1 + (-10.9 + 6.31i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.08T + 97T^{2} \)
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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

−9.710778246177307976796635400765, −9.105187799708506589026054471260, −8.318515399892012902713607812130, −7.26412556965526461205094109262, −6.75789135910380806493616232297, −5.62540245481610396789402720398, −4.96288299906987693847501145158, −4.07338305402839559570529800076, −2.29908636703836337710636035180, −0.805239803229678138187522588629, 1.91824481197374694372744729072, 3.14134911668594920698333680545, 3.81389711430981188029301376277, 4.94868967993703417969907586134, 5.88039784713839499171075777586, 6.63469515733377848586331049180, 7.81029780959038920634009309375, 9.300654071033802327872525409945, 10.15741367623916658522891290441, 10.36430343268188712124569861672

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